2022

1]Guangxi Key Laboratory for Relativistic Astrophysics, School of Physical
Science and Technology, Guangxi University, Nanning 530004, China 2]INAF
Istituto di Astrofisica e Planetologia Spaziali, Via del Fosso del Cavaliere
100, 00133 Roma, Italy 3]INAF Osservatorio Astrofisico di Arcetri, Largo
Enrico Fermi 5, 50125 Firenze, Italy 4]Dipartimento di Fisica e Astronomia,
Università degli Studi di Firenze, Via Sansone 1, 50019 Sesto Fiorentino (FI),
Italy 5]Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, Via Sansone
1, 50019 Sesto Fiorentino (FI), Italy 6]Department of Physics and Kavli
Institute for Particle Astrophysics and Cosmology, Stanford University,
Stanford, California 94305, USA 7]INAF Osservatorio Astronomico di Cagliari,
Via della Scienza 5, 09047 Selargius (CA), Italy 8]Yamagata University,1-4-12
Kojirakawa-machi, Yamagata-shi 990-8560, Japan 9]Istituto Nazionale di Fisica
Nucleare, Sezione di Torino, Via Pietro Giuria 1, 10125 Torino, Italy 10]NASA
Goddard Space Flight Center; Greenbelt, MD 20771, USA 11]University of
Maryland, Baltimore County, Baltimore, MD 21250, USA 12]Center for Research
and Exploration in Space Science and Technology, NASA/GSFC, Greenbelt, MD
20771, USA 13]Instituto de Astrofísica de Andalucía—CSIC, Glorieta de la
Astronomía s/n, 18008, Granada, Spain 14]INAF Osservatorio Astronomico di
Roma, Via Frascati 33, 00040 Monte Porzio Catone (RM), Italy 15]Space Science
Data Center, Agenzia Spaziale Italiana, Via del Politecnico snc, 00133 Roma,
Italy 16]Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Largo B.
Pontecorvo 3, 56127 Pisa, Italy 17]Dipartimento di Fisica, Università di Pisa,
Largo B. Pontecorvo 3, 56127 Pisa, Italy 18]NASA Marshall Space Flight Center,
Huntsville, AL 35812, USA 19]Dipartimento di Matematica e Fisica, Università
degli Studi Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy
20]Dipartimento di Fisica, Università degli Studi di Torino, Via Pietro Giuria
1, 10125 Torino, Italy 21]Agenzia Spaziale Italiana, Via del Politecnico snc,
00133 Roma, Italy 22]Istituto Nazionale di Fisica Nucleare, Sezione di Roma
Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy 23]Institut
für Astronomie und Astrophysik, Sand 1, 72076 Tübingen, Germany
24]Astronomical Institute of the Czech Academy of Sciences, Boční II 1401/1,
14100 Praha 4, Czech Republic 25]RIKEN Cluster for Pioneering Research, 2-1
Hirosawa, Wako, Saitama 351-0198, Japan 26]California Institute of Technology,
Pasadena, CA 91125, USA 27]Osaka University, 1-1 Yamadaoka, Suita, Osaka
565-0871, Japan 28]University of British Columbia, Vancouver, BC V6T 1Z4,
Canada 29]Department of Physics, Faculty of Science and Engineering, Chuo
University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan 30]Institute for
Astrophysical Research, Boston University, 725 Commonwealth Avenue, Boston, MA
02215, USA 31]Laboratory of Observational Astrophysics, St. Petersburg
University, University Embankment 7/9, St. Petersburg 199034, Russia
32]Physics Department and McDonnell Center for the Space Sciences, Washington
University in St. Louis, St. Louis, MO 63130, USA 33]Finnish Centre for
Astronomy with ESO, Vesilinnantie 5, 20014 University of Turku, Finland
34]Université de Strasbourg, CNRS, Observatoire Astronomique de Strasbourg,
UMR 7550, 67000 Strasbourg, France 35]MIT Kavli Institute for Astrophysics and
Space Research, Massachusetts Institute of Technology, 77 Massachusetts
Avenue, Cambridge, MA 02139, USA 36]Graduate School of Science, Division of
Particle and Astrophysical Science, Nagoya University, Furo-cho, Chikusa-ku,
Nagoya, Aichi 464-8602, Japan 37]Hiroshima Astrophysical Science Center,
Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526,
Japan 38]Department of Physics, University of Hong Kong, Pokfulam, Hong Kong
39]Department of Astronomy and Astrophysics, Pennsylvania State University,
University Park, PA 16801, USA 40]Université Grenoble Alpes, CNRS, IPAG, 38000
Grenoble, France 41]Department of Physics and Astronomy, University of Turku,
20014 Turku, Finland 42]Space Research Institute of the Russian Academy of
Sciences, Profsoyuznaya Str. 84/32, Moscow 117997, Russia 43]Center for
Astrophysics, Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA
44]INAF Osservatorio Astronomico di Brera, via E. Bianchi 46, 23807 Merate
(LC), Italy 45]Dipartimento di Fisica e Astronomia, Università degli Studi di
Padova, Via Marzolo 8, 35131 Padova, Italy 46]Dipartimento di Fisica,
Università degli Studi di Roma Tor Vergata, Via della Ricerca Scientifica,
00133 Roma, Italy 47]Mullard Space Science Laboratory, University College
London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK 48]Anton Pannekoek
Institute for Astronomy & GRAPPA, University of Amsterdam, Science Park 904,
1098 XH Amsterdam, The Netherlands

# Vela pulsar wind nebula x-rays are polarized to near the synchrotron limit

Fei Xie1,2 Corresponding author<EMAIL_ADDRESS>Alessandro Di Marco2 Fabio La
Monaca2 Kuan Liu1 Fabio Muleri2 Niccolò Bucciantini3,4,5 Roger W. Romani6
Enrico Costa2 John Rankin2 Paolo Soffitta2 Matteo Bachetti7 Niccolò Di Lalla6
Sergio Fabiani2 Riccardo Ferrazzoli2 Shuichi Gunji8 Luca Latronico9 Michela
Negro10,11,12 Nicola Omodei6 Maura Pilia7 Alessio Trois7 Eri Watanabe8 Iván
Agudo13 Lucio A. Antonelli14,15 Luca Baldini16,17 Wayne H. Baumgartner18
Ronaldo Bellazzini16 Stefano Bianchi19 Stephen D. Bongiorno18 Raffaella
Bonino9,20 Alessandro Brez16 Fiamma Capitanio2 Simone Castellano16 Elisabetta
Cavazzuti21 Stefano Ciprini15,22 Alessandra De Rosa2 Ettore Del Monte2 Laura
Di Gesu21 Immacolata Donnarumma21 Victor Doroshenko23 Michal Dovčiak24 Steven
R. Ehlert18 Teruaki Enoto25 Yuri Evangelista2 Javier A. Garcia26 Kiyoshi
Hayashida27,† Jeremy Heyl28 Wataru Iwakiri29 Svetlana G. Jorstad30,31 Vladimir
Karas24 Takao Kitaguchi25 Jeffery J. Kolodziejczak18 Henric Krawczynski32
Ioannis Liodakis33 Simone Maldera9 Alberto Manfreda16 Frédéric Marin34 Andrea
Marinucci21 Alan P. Marscher30 Herman L. Marshall35 Francesco Massaro9,20
Giorgio Matt19 Ikuyuki Mitsuishi36 Tsunefumi Mizuno37 C.-Y. Ng38 Stephen L.
O’Dell18 Chiara Oppedisano9 Alessandro Papitto14 George G. Pavlov39 Abel L.
Peirson6 Matteo Perri14,15 Melissa Pesce-Rollins16 Pierre-Olivier Petrucci40
Andrea Possenti7 Juri Poutanen41,42 Simonetta Puccetti21 Brian D. Ramsey18
Ajay Ratheesh2 Carmelo Sgró16 Patrick Slane43 Gloria Spandre16 Toru Tamagawa25
Fabrizio Tavecchio44 Roberto Taverna45 Yuzuru Tawara36 Allyn F. Tennant18
Nicolas E. Thomas18 Francesco Tombesi46 Sergey S. Tsygankov41,42 Roberto
Turolla45,47 Jacco Vink48 Martin C. Weisskopf18 Kinwah Wu47 Silvia Zane47
*Corresponding author<EMAIL_ADDRESS>[  [  [  [  [  [  [  [  [  [  [  [  [
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Pulsar wind nebulae are formed when outflows of relativistic electrons and
positrons hit the surrounding supernova remnant or interstellar medium at a
shock front. The Vela pulsar wind nebula is powered by a young pulsar
(B0833-45, age 11 kyr) 1968Natur.220..340L  and located inside an extended
structure called Vela X, itself inside the supernova remnant 1985ApJ…299..828H
. Previous X-ray observations revealed two prominent arcs, bisected by a jet
and counter jet 2001ApJ…554L.189P ; 2001ApJ…556..380H . Radio maps have shown
high linear polarization of 60 per cent in the outer regions of the nebula
2003MNRAS.343..116D . Here we report X-ray observation of the inner part of
the nebula, where polarization can exceed 60 per cent at the leading edge,
which approaches the theoretical limit of what can be produced by synchrotron
emission. We infer that, in contrast with the case of the supernova remnant,
the electrons in the pulsar wind nebula are accelerated with little or no
turbulence in a highly uniform magnetic field.

Imaging X-ray Polarimetry Explorer (IXPE) 2021AJ….162..208S ;
2022JATIS…8b6002W  provides a new X-ray view of the highest energy electrons
near their acceleration sites. IXPE is a NASA/ASI explorer featuring three co-
aligned X-ray telescopes each with an imaging photoelectric polarimeter
detector unit based on a gas pixel detector Costa2001 ; 2021APh…13302628B .
IXPE observed the Vela pulsar wind nebula (PWN) in two periods: (i) 2022-04-05
to 2022-04-15 and (ii) 2022-04-21 to 2022-04-30, with a total exposure of 860
ks. The data were extracted from the publicly available files processed by the
IXPE Science Operations Center and analysed with standard tools as described
in the supplementary Methods section.

Linear polarization is detected at high significance ($\sim$ 31$\sigma$) for
the spatial- and energy-integrated Vela PWN in the 2–8 keV band. The linear
polarization degree (PD) is (44.6$\pm$1.4)% with a polarization angle (PA,
also known as electric vector position angle EVPA, defined from North through
East) of $-(50.0\pm$0.9)∘, with 68.3% confidence level uncertainties.
Previously, only the Crab PWN has been studied via X-ray polarization; the
classic 1976, 1978 OSO-8 results 1976ApJ…208L.125W ; 1978ApJ…220L.117W  have
recently been confirmed by the Polarlight cube-sat 2020NatAs…4..511F .
However, these observations provided only an integrated polarization
measurement and the image-averaged polarization degree was less than half of
that found here for Vela.

IXPE’s imaging capabilities, with a $\leq 30^{\prime\prime}$ half power
diameter 2022JATIS…8b6002W , allow a spatially resolved polarimetric
measurement of the PWN (Fig. 1). Here, 25 independent
30${}^{\prime\prime}\times$30′′ square regions have been analyzed, combining
data from the 3 detector units. The pulsar and compact arc-jet structure lie
largely within the central region. The pulsar, dominated by a $kT\sim 0.13$
keV thermal component 2001ApJ…552L.129P , contributes less than 10% of the
IXPE counts in this region. IXPE has not measured the polarization of the
pulsed source; this may add to or subtract from the nebular polarization, but
with the low flux (and low expected polarization degree Bucciantini et al )
the effect should be small. The black lines show linear polarization, with
line length indicating polarization degree up to 62.8%. These lines are at 90∘
to the EVPA, and thus show the direction of the projected magnetic field,
which is highly symmetric about the pulsar jet axis.

The curved and symmetric polarization angle pattern seen in Fig. 1 implies
that there will be some polarization angle variation across, and decrease in
the region-average polarization degree of, our 25 measurement regions. This is
most obvious to the sides and rear of the PWN, where the arcs seen in the
Chandra image are highly curved and thus the underlying polarization angle
varies rapidly, producing substantial de-polarization. Since the polarization
follows the Chandra X-ray morphology we expect slightly higher polarization
degree and smaller errors in regions where the arcs are less curved, which
allows us to average over larger areas. As an example a morphologically
selected region at the front of the nebula, along the symmetry axis provides a
PD=70.0$\pm$3.6% (Methods and Extended Data Fig. 5). As a caution, we note
that for an analysis near the angular resolution limit in regions covering
sharp intensity gradients, an effect due to the reconstruction of the photo-
electron tracks in the gas pixel detector can artificially alter the local
polarization values. Monte Carlo analysis show that this is a $\lesssim 5$%
effect, with a slight decrease in polarization degree to the North and
increase to the South; our maximally polarized zones are hardly affected.
Regardless, Vela’s high polarization degree and symmetric pattern imply a
highly ordered magnetic field following the PWN’s toroidal structure.

The spatially-averaged polarization of the Vela PWN is plotted as a function
of energy in Fig. 2. A slight increase of polarization degree with energy is
seen. This might be expected, since the emission region shrinks with
increasing energy, thus arising from a smaller range of magnetic field
orientations. The image-averaged polarization angle does not show significant
energy dependence.

It is widely accepted that PWN arc-jet structures are due to an anisotropic
pulsar wind, which generates toroidal magnetic fields in the equatorial zone
2004ApJ…601..479N ; 2002MNRAS.329L..34L ; 2003MNRAS.344L..93K ;
2004A&A…421.1063D . In this picture, the integrated polarization angle should
align with the symmetry axis of the Vela PWN 2004MNRAS.349..779K ;
2005A&A…443..519B ; 2006A&A…453..621D , well consistent with the results
reported here. The non-thermal radio and X-ray spectra of the Vela PWN already
indicated domination by synchrotron emission. The high linear X-ray
polarization, measured here for the first time with IXPE, strengthens this
conclusion.

The maximum polarization degree of synchrotron radiation from a power law
spectrum of electrons with index $p$ in a uniform magnetic field is
$\Pi=(p+1)/(p+7/3)$ 1986rpa..book…..R ; the photon index $\Gamma$ of the
emitted radiation is given by $p=2\Gamma-1$. For Vela, with $\Gamma=1.3\pm
0.04$ in the compact jet-arc structure, increasing to 1.7 in the outskirts
2002ASPC..271..181K , the maximum polarization degree is 66–72%. Thus the
62.8% seen in Fig. 1 and Extended Data Table 3 (or the 70% in regions selected
according to the Chandra image in Extended Data Table 4) is quite close to the
maximum permitted value. This implies a magnetic field that is highly uniform
across the measured region, with little turbulence-induced fluctuations. Also
$\Gamma=1.3$ implies $p=1.6$, appreciably flatter than the $p\approx 2.3-2.6$
of turbulent diffusive shock acceleration 2015SSRv..191..519S ; another
mechanism, such as reconnection 2014ApJ…783L..21S , should play a dominant
role in PWN particle energization.

The high uniformity and toroidal magnetic structure that we see with IXPE
evidently extends to larger radii, where the $4^{\prime}$ radio lobes’ linear
polarization indicates a magnetic field with a similar symmetry axis
2003MNRAS.343..116D  (Fig. 1). Interestingly this magnetic structure is
compressed to the North (evidently by the PWN interaction with the larger
supernova remnant) and has a larger radius of curvature, centered behind the
pulsar. With little radio emission from the X-ray bright torus/jet zone in the
center, it appears that the radio tracks an older, cooler electron population
centered behind the pulsar and extending to larger radii. Still, the
substantial radio polarization, reaching 60% at both 5-GHz 2003MNRAS.343..116D
and 1.4-GHz 2002ASPC..271..187B , indicates that the ordered and minimally
turbulent fields probed by IXPE extend into this radio-emitting zone at much
larger radii.

At present, we can say little about the X-ray polarization from the pulsar
itself. Much deeper observations (and likely higher spatial resolution) will
be required to isolate this component and compare with the optical phase
average polarization of PD=8.1$\pm 0.6$% at
PA=146.3$\pm$2.4∘2014MNRAS.445..835M . Like the Crab pulsar, Vela’s average
optical polarization angle lies close to the projected torus axis. If, like
the Crab pulsar, the phase-average X-ray polarization degree is well below
that of the optical emission, corrections to our nebular estimates will be
very small.

We find a remarkable high X-ray polarization in the Vela PWN, reaching an
image- and energy-averaged polarization degree of $\sim$ 45%. The non-thermal
PWN spectrum is bright in the IXPE energy band; the Vela pulsar itself emits
mostly soft thermal X-rays, too faint and weakly polarized for detection at
present. Our IXPE image sufficiently resolves the PWN to show that the
polarization structure is symmetric about the projected pulsar spin (and
proper motion) axis; this symmetry extends in radio polarization studies to
even larger angles. The varying polarization angle across the nebula implies
that the true local polarization is even larger and indeed we find
polarization degree values $\geq$ 60% for some regions. This is close to the
maximum polarization degree allowed for synchrotron emission with the observed
X-ray spectrum, implying that the magnetic field is highly uniform across the
emission region. Since electrons emitting synchrotron X-rays cool very
rapidly, the IXPE X-rays are emitted close to the acceleration zone. In turn
this argues against turbulence-driven diffusive shock acceleration and
suggests that other processes, such as reconnection, should energize the PWN
particles in the termination shock. Further IXPE studies of Vela and other
bright PWNe should connect the X-ray polarization pattern with the details of
the compact structures, further probing the physics of relativistic shock
acceleration.

## References

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Figure 1:  The IXPE intensity map of the Vela PWN in the 2–8 keV range with
the measured X-ray polarization and radio polarization vectors overlaid.
Intensity is the Gaussian-smoothed sum of the three detector units. The black
lines present the X-ray polarization in the corresponding
30${}^{\prime\prime}\times$30′′ image elements (white grids), their lengths
indicate the polarization degree and the orientation indicates the projected
magnetic field ($90^{\circ}$ from the EVPA). The thinner silver lines, drawn
with the same lengths, show the 5 GHz polarization directions derived from
ref. 2003MNRAS.343..116D . The gray contours, obtained from Chandra
observations in the same 2–8 keV range, give a better illustration of the
pulsar, compact arc-jet structure, diffuse emission shell, and relatively weak
outer jet (only weakly detected at IXPE resolution). The bar at the bottom
left presents the maximum measured X-ray polarization degree of 63%, and the
PD and PA in each of 25 grids are tabulated in Extended Data Table 3.

Figure 2: Image-average polarization of the Vela PWN as a function of energy.
Polarization degree (top) and polarization angle (bottom) are derived with
ixpeobssim (red) and xspec (blue) independently. The results are from the
joint analysis of three detector units with uncertainties calculated for a
68.3% confidence level. Polarization degrees in all energy bins are
significantly higher than the minimum detectable polarization at the 99%
confidence level (MDP99) shown as a gray histogram.

## Methods

### IXPE Data Analysis

This work is based on the polarimetric observations of the Vela pulsar and
nebula obtained with the Imaging X-ray Polarimetry Explorer (IXPE), a NASA
mission in partnership with the Italian space agency (ASI), as described in
ref. 2021AJ….162..208S ; 2022JATIS…8b6002W  and references therein. The IXPE
Observatory includes three identical X-ray telescopes, each comprising an
X-ray mirror assembly and a linear-polarization-sensitive detector, and
performs imaging polarimetry over a nominal 2–8 keV energy band. IXPE data are
telemetered to ground stations in Malindi (primary) and in Singapore
(secondary), and are transmitted to the NASA Marshall Space Flight Center
Science Operations Center (SOC) via the Mission Operations Center (MOC), at
the Laboratory for Atmospheric and Space Physics, University of Colorado. Data
are processed at the SOC, where raw data and relevant engineering/ancillary
data are used to produce photon event lists. For each observation, the data
are archived at the High-Energy Astrophysics Science Archive Research Center
(HEASARC), at the NASA Goddard Space Flight Center for use by the
international astrophysics community.

IXPE data are provided in two levels/formats: event_l1 and event_l2. The first
level consists of unfiltered events; event_l2 files are produced from event_l1
after further calibration/correction procedures. In particular, event_l2 data
are cleaned of in-flight-calibration data and occultation/SAA passages, then
energy calibration/equalization of the three detector units (DUs) is applied
and Stokes parameters are estimated following ref. 2015APh….68…45K . After
this the spurious modulation is removed Rankin_2022  and the photons are
referenced to sky coordinates, removing dithering pattern and boom motion
effects.

The Vela pulsar and PWN were observed with IXPE from 2022-04-05 at
19:51:40.687 UTC to 2022-04-15 at 18:08:18.343 UTC, and again from 2022-04-21
at 12:22:11.549 UTC to 2022-04-30 at 10:34:51.991 UTC, with net exposures 860
ks, 854 ks, 868 ks, respectively for the three DUs. After retrieving event_l2
files from the HEASARC, we performed three corrections: (1) energy correction
(based on the onboard calibration data); (2) bad time interval filtering (to
remove the solar flare events, see Extended Data Fig. 1); (3) barycenter
correction (to convert the photon arrival time into the solar system
barycenter; done with the standard HeaSoft 6.30.1 barycorr, with the following
options: refframe=ICRS, ephem=JPLEPH.421). We find that the IXPE Vela
observations were slightly affected by an increased particle flux during some
portions of the orbit, associated with high solar activity.

Bad time intervals are identified as periods with count rates 3$\sigma$ higher
than average, assuming a Gaussian distribution. Events during such bad
intervals are excluded from further data analysis.

The field of view (FoV) of each detector is 12.9${}^{\prime}\times$ 12.9′
(with the three detectors mounted at angles of 120∘ to each other)
2022JATIS…8b6002W , while the X-ray emission of Vela PWN covers a much
smaller region 2001ApJ…554L.189P . We identified source and background
regions using the SAOImageDS9 software 2006ASPC..351..574J . The source
region is a circle of radius 1′ centered on the pulsar. The background region
is an annulus with inner radius of 2′ and outer radius of 4.7′ (see Extended
Data Fig. 2).

The polarization data consists of Stokes parameters ($I$, $Q$, $U$) for each
photon in the event_l2 files. For details on the Stokes parameters as defined
in IXPE data see ref. Di_Marco_2022 . Data analysis was performed with
different techniques and by different groups within the IXPE collaboration to
cross-check the results. We first analyzed the data with the IXPE internal
software ixpeobssim 2022arXiv220306384B , customized for IXPE data analysis
and simulations. ixpeobssim includes tools developed to determine the source
polarization properties via an analysis independent of the spectral models,
following the unbinned method described in ref. 2015APh….68…45K . A second
reduction is performed with xspec, using the version heasoft 6.30.1, which
includes models for spectro-polarimetric analysis 2017ApJ…838…72S . With the
binned spectra of the Stokes parameters, xspec performs standard forward-
fitting and derives model-dependent polarization parameters. The adopted IXPE
response functions are from the HEASARC IXPE CALDB March 14th, 2022 release.

The spectral analysis shows that background events are not negligible at high
energies, with rates becoming comparable (see Extended Data Fig. 3). To
determine the source polarization while avoiding dilution effects at high
energies, the background component was subtracted in all the analyses. The 2–8
keV band-averaged, aperture-averaged (1′ radius) polarization degree and angle
measurements are summarized in Extended Data Table
LABEL:tab:spec_pol_energy_bin. The results are consistent between the three
DUs and the two different analysis methods.

Spectro-polarimetric analysis is performed using xspec 1996ASPC..101…17A  to
jointly fit the three DUs in a two-step procedure. In the first step, the $I$
energy distribution are fitted with a spectral model. In the second step the
spectral model is fixed, while $U$ and $Q$ are fit. This method thus does a
forward folding fit to the binned spectra of the Stokes parameters $I$, $Q$,
and $U$ jointly after fixing the spectral model.

The Vela PWN binned $I$ spectra from the three DUs were fitted with the model
factor*tbabs*powerlaw, where tbabs takes into account the interstellar
absorption – here, we fixed the column density 2001ApJ…552L.129P  to
$N_{H}=0.03\times 10^{22}$ cm-2 – a relative normalization factor for DU2 and
DU3 accounts for uncertainties in their absolute effective area. The best-fit
curves and the spectra of the Vela PWN for the three DUs are shown in Extended
Data Fig. 3. At present, uncertainties in the effective area energy dependence
are substantial, due to the incomplete knowledge of telescope alignment and
related vignetting. Accordingly we focus on the polarization information. The
spectral model parameters do not impact the polarization results, which are
affected only by the energy assigned to individual photons. The individual
photon energy estimate is tied to frequent in-flight calibration, performed
with onboard calibration source during the Earth occultation. As the effective
area variation becomes better understood, we expect the spectral model fitting
to improve.

With the spectral model, binned $I$, $Q$, and $U$ spectra are fitted to obtain
the polarimetric information, using the polarization model polconst of xspec.
The fitted $I$, $Q$, $U$ spectra in the 2–8 keV energy band are shown in
Extended Data Fig. 3. The polarization angle and degree obtained are in good
agreement with those obtained by the model-independent analysis performed with
ixpeobssim, as reported in Extended Data Table LABEL:tab:spec_pol_energy_bin.

The same procedure was used to investigate the polarization in six energy bins
fitting with xspec model factor*tbabs*powerlaw*polconst in each energy
interval, freezing the $I$ spectral model parameters to the values previously
obtained from the 2–8 keV analysis. Polarization degree and angle in each
energy bin are reported in Extended Data Table LABEL:tab:spec_pol_energy_bin
and Fig. 2. The ixpeobssim analysis is fully in agreement with that of xspec.
PD and PA values obtained with xspec are represented in a 2D polar plot – with
contours enclosing the 68.3% C.L. regions – in Extended Data Fig. 4.
Similarly, $Q/I$ and $U/I$ Stokes parameters obtained with xspec and
ixpeobssim are shown in Extended Data Fig. 6.

We have attempted to quantify the significance of the PD$(E)$ trend in Fig. 2.
While a $\chi^{2}$ test finds that a constant PD is acceptable at the 57%
level, a run-test analysis only accepts this hypothesis at 7.7% and a
Kolmogorov-Smirnov test only has a probability of 3.6%. In contrast a linear
PD$(E)$ model gives an acceptable $\chi^{2}$ probability at the 98% level, the
run-test statistic is allowed at 41%, and a K-S test accepts the hypothesis at
the 98% level. Thus a linear model with slope (1.8$\pm$0.9)$\times 10^{-2}$
keV-1, while not demanded by the data, is preferred over a simple constant PD
model at the 2$\sigma$ confidence level.

#### Spatially-resolved Analysis

Spatially-resolved analysis is performed with ixpeobssim. We divided the FoV
into 30′′ square bins, guided by the mirror PSF. Polarization analysis was
performed for each bin, without background subtraction. Removing the
background counts would give a small increase in the PD without affecting the
PA. The statistics in the outer bins are poor, so we present only the results
in 5$\times$5 grids centered on the pulsar (Fig. 1). The tabulated results of
the normalized $Q$, $U$, and their significance are in Extended Data Table 2,
and the corresponding PD and PA are presented in Extended Data Table 3.

We also analyze the polarization parameters in five larger regions, aligned to
the PWN symmetry axis, labeled as $L$, $R$, $F$, $B$ and $C$ in Extended Data
Fig. 5. Here we use ixpeobssim and subtract a scaled off-source background.
The results, tabulated in Extended Data Table 4, show a polarization degree
increasing along the symmetry axis from $\sim 30\%$ ($B$ region) up to $\sim
70\%$ ($F$ region). As with the finer square grid, $L$ and $R$ side regions
show PA symmetrically spread about PA for the symmetry axis region $C$.

### The effect of the Pulsar on the PWN Polarization

The 89 ms Vela pulsar’s X-rays are largely thermal and make a negligible
contribution to the 2–8 keV PWN flux as a whole. Thus, they cannot
significantly affect our image-averaged PD and PA. However, from high
resolution Chandra images, we see that the pulsar contributes $\sim 10\%$ of
the 2–8 keV flux in the central square region of Fig. 1 and so it might affect
its polarization. To bound such effects, we measured the spectrum of the Vela
pulsar using the Chandra observation obsid 218, processed through the standard
pipeline with the CIAO 4.13 package and CALDB version 4.9.5. If the pulsar
contributes an unpolarized background, the corrected nebula-only PD rises to
55%. If in contrasts the pulsar emission is 100% polarized parallel
(perpendicular) to the PA measured for the central square region, then the
pulsar-subtracted polarization falls (rises) to 39% (66%). These values are
from DU1 measurements only since with the smallest half-power width the pulsar
will be most severe for this DU. In practice, the pulsar polarization is quite
low in the optical band (PD=8.1$\pm$0.6% at PA=(146.3$\pm$0.6)∘)
2014MNRAS.445..835M , and, by analogy with the Crab, will be even lower in
the X-rays, so the true effect should be close to the unpolarized case,
boosting the inferred PWN central PD. We apply this correction to the central
bin in Fig. 1. Follow-up IXPE studies may more tightly bound the polarization
of the pulsed emission.

## Data Availability

IXPE data are available through the NASA’s HEASARC data archive at
https://heasarc.gsfc.nasa.gov. Other derived data, supporting the findings of
this study, are available from the corresponding author upon request.

## Code Availability

The High Energy Astrophysics Science Archive Research Center (HEASARC)
developed the HEASOFT (HEASARC Software). We used the HEASOFT version 6.30.1
package for the spectro-polarimetric IXPE data analysis, available at:
https://heasarc.gsfc.nasa.gov/docs/software/heasoft/. The proper instrument
response functions are provided by the IXPE Team as a part of the IXPE
calibration database released on 2022 March 14 and available in the HEASARC
Calibration Database at:
https://heasarc.gsfc.nasa.gov/docs/heasarc/caldb/caldb_supported_missions.html.

The software developed by IXPE collaboration is available publicly through the
web-page https://ixpeobssim.readthedocs.io/en/latest/?badge=latest.

Information supporting the findings of this study is available from the
corresponding author upon request.

## References

  *   * (1) Kislat, F., Clark, B., Beilicke, M. & Krawczynski, H.  Analyzing the data from X-ray polarimeters with Stokes parameters.  _Astropart. Phys._ 68, 45–51 (2015).
  * (2) Rankin, J. _et al._ An Algorithm to Calibrate and Correct the Response to Unpolarized Radiation of the X-Ray Polarimeter Onboard IXPE.  _Astron. J._ 163, 39 (2022).
  * (3) Joye, W. A.  New Features of SAOImage DS9.  In _Astronomical Data Analysis Software and Systems XV_ , vol. 351 of _Astronomical Society of the Pacific Conference Series_ , 574 (2006).
  * (4) Di Marco, A. _et al._ A Weighted Analysis to Improve the X-Ray Polarization Sensitivity of the Imaging X-ray Polarimetry Explorer.  _Astron. J._ 163, 170 (2022).
  * (5) Baldini, L. _et al._ ixpeobssim: a Simulation and Analysis Framework for the Imaging X-ray Polarimetry Explorer.  _arXiv_ (2022).
  * (6) Strohmayer, T. E.  X-Ray Spectro-polarimetry with Photoelectric Polarimeters.  _Astrophys. J._ 838, 72 (2017).
  * (7) Arnaud, K. A.  XSPEC: The First Ten Years.  In _Astronomical Data Analysis Software and Systems V_ , vol. 101 of _Astronomical Society of the Pacific Conference Series_ , 17 (1996).

## Acknowledgements

The Imaging X ray Polarimetry Explorer (IXPE) is a joint US and Italian
mission. The US contribution is supported by the National Aeronautics and
Space Administration (NASA) and led and managed by its Marshall Space Flight
Center (MSFC), with industry partner Ball Aerospace (contract NNM15AA18C). The
Italian contribution is supported by the Italian Space Agency (Agenzia
Spaziale Italiana, ASI) through contract ASI-OHBI-2017-12-I.0, agreements ASI-
INAF-2017-12-H0 and ASI-INFN-2017.13-H0, and its Space Science Data Center
(SSDC) with agreements ASI-INAF-2022-14-HH.0 and ASI-INFN 2021-43-HH.0, and by
the Istituto Nazionale di Astrofisica (INAF) and the Istituto Nazionale di
Fisica Nucleare (INFN) in Italy. This research used data products provided by
the IXPE Team (MSFC, SSDC, INAF, and INFN) and distributed with additional
software tools by the High-Energy Astrophysics Science Archive Research Center
(HEASARC), at NASA Goddard Space Flight Center (GSFC). The research at Guangxi
University was supported in part by National Natural Science Foundation of
China (Grant No. 12133003). The research at Boston University was supported in
part by National Science Foundation grant AST-2108622. I.A. acknowledges
financial support from the Spanish “Ministerio de Ciencia e Innovación”
(MCINN) through the “Center of Excellence Severo Ochoa” award for the
Instituto de Astrofísica de Andalucía-CSIC (SEV-2017-0709) and through grants
AYA2016-80889-P and PID2019-107847RB-C44.

## Author contributions

F.X. led the data analysis and the writing of the paper. A.D.M. and F.L.M
performed the spectro-polarimetric analysis and contributed to the manuscript.
F.M. and J.R. contributed to data analysis and energy calibration of the data-
set. K.L. and N.B. contributed to the data analysis and results
interpretation. R.W.R. and L.L. helped revise the manuscript. E.C., P.S.,
M.B., S.F., R.F., M.P. and A.T. contributed to the interpretation of the
results and to text revisions. N.D.L., S.G., N.O., M.N. and E.W. performed
independent analysis of the data. The remaining members of the IXPE
collaboration contributed to the design of the mission, to the calibration of
the instrument, to definition its science case and to the planning of the
observations. All authors provided inputs and comments on the manuscript.

## Competing interests

The authors declare no conflicts of interest.

## Extended Data

Extended Data Fig. 1: Light curve of the Vela PWN observed by IXPE DU1 before
and after data filtering (top and bottom panels, respectively). Filtering the
high count-rate excursions from the event_l2 file removes $\sim 1.47$% of the
exposure time.

Extended Data Fig. 2: Total nebula source (white 1′ radius circle) and
background (blue annulus, inner radius 2′, outer radius 4.7′) regions shown on
images from DU1 (left) to DU3 (right). Intensity is on a logarithmic scale to
bring out the faint background.

Extended Data Fig. 3: Spectral joint fitting for the Stokes parameters in the
2–8 keV energy band for the three DUs using model tbabs*powerlaw*polconst with
previously fit spectral parameters fixed. The average background count $I$
spectrum for the three DUs is shown in black. Fit residuals are shown at the
bottom.

Extended Data Fig. 4: Polar plot showing the polarization degree (PD) and
polarization angle (PA) fit with data from the three DUs, for different energy
bands. Ellipses show the 68.3% confidence level errors obtained with xspec.

Extended Data Fig. 5: An alternative spatial partition of the PWN with regions
aligned with the PWN symmetry axis on image observed by Chandra. $L$, $R$,
$F$, $B$ label the left, right, front, back regions with respect to the $C$
center region respectively, corresponding to the analysis tabulated in
Extended Data Table 4. The inner circle has a radius of 15′′ and the outer
circle radius is 45′′.

Extended Data Fig. 6: $Q/I$ (top) and $U/I$ (bottom) Stokes parameters of the
Vela PWN as functions of energy in 2–8 keV range derived with ixpeobssim and
xspec independently. The results are from the joint analysis of three DUs.

Extended Data Table 1: Polarization degree and angle in six different energy
bands and in the full IXPE 2–8 keV band.

| 2–3 keV | 3–4 keV | 4–5 keV | 5–6 keV | 6–7 keV | 7–8 keV | 2–8 keV
---|---|---|---|---|---|---|---
PD (%)I | 42.3$\pm$1.8 | 43.8$\pm$1.9 | 45.2$\pm$2.7 | 46.4$\pm$3.8 | 50.1$\pm$5.7 | 56.1$\pm$9.7 | 44.6$\pm$1.4
PD (%)X | 43.4$\pm$1.9 | 45.4$\pm$2.0 | 46.0$\pm$2.9 | 47.4$\pm$4.1 | 46.9$\pm$5.7 | 50.2$\pm$9.2 | 44.9$\pm$1.1
PA (∘)I |  $-$49.9$\pm$1.2 |  $-$50.0$\pm$1.2 |  $-$49.3$\pm$1.7 |  $-$49.7$\pm$2.4 |  $-$51.7$\pm$3.3 |  $-$54.7$\pm$5.0 |  $-$50.0$\pm$0.9
PA (∘)X |  $-$49.9$\pm$1.2 |  $-$50.0$\pm$1.3 |  $-$49.4$\pm$1.8 |  $-$50.0$\pm$2.5 |  $-$51.2$\pm$3.5 |  $-$51.1$\pm$5.4 |  $-$50.0$\pm$0.7

  * Polarization degree and polarization angle measured using ixpeobssim and xspec in six energy bands and 2–8 keV using three DUs. Uncertainties on the polarization degree and angle are calculated for a 68.3% confidence level, assuming that they are independent. The values from the two different methods are compatible within the uncertainties.

  * I

Values are obtained with ixpeobssim.

  * X

Values are obtained with xspec.

Extended Data Table 2: The normalized Stokes parameters and the detection
significance within the bins of Fig. 1.

| -2b | -1b | 0b | 1b | 2b |
---|---|---|---|---|---|---
|  $0.33\pm$0.17 |  $0.20\pm$0.13 |  $0.07\pm$0.11 |  $-0.09\pm$0.13 |  $-0.14\pm$0.15 | Q/Ic
2a |  $-0.18\pm$0.18 |  $-0.18\pm$0.13 |  $-0.61\pm$0.11 |  $-0.36\pm$0.13 |  $-0.45\pm$0.14 | U/Id
| $1.3$ | $1.2$ | $5.0$ | $2.0$ | $2.6$ | Sige
|  $0.32\pm$0.10 |  $0.34\pm$0.05 |  $0.05\pm$0.04 |  $-0.10\pm$0.07 |  $-0.21\pm$0.13 | Q/Ic
1a |  $0.07\pm$0.10 |  $-0.34\pm$0.05 |  $-0.53\pm$0.04 |  $-0.56\pm$0.07 |  $-0.42\pm$0.13 | U/Id
| $2.5$ | $9.7$ | $12.9$ | $7.6$ | $3.1$ | Sige
|  $0.10\pm$0.09 |  $0.13\pm$0.04 |  $-0.09\pm$0.02 |  $-0.19\pm$0.04 |  $-0.08\pm$0.11 | Q/Ic
0a |  $-0.03\pm$0.09 |  $-0.32\pm$0.04 |  $-0.48\pm$0.02 |  $-0.60\pm$0.04 |  $-0.43\pm$0.11 | U/Id
| $-$ | $8.8$ | $19.7$ | $15.6$ | $3.3$ | Sige
|  $-0.01\pm$0.12 |  $-0.20\pm$0.07 |  $-0.30\pm$0.04 |  $-0.44\pm$0.05 |  $-0.18\pm$0.12 | Q/Ic
-1a |  $-0.20\pm$0.12 |  $-0.19\pm$0.07 |  $-0.53\pm$0.04 |  $-0.37\pm$0.05 |  $-0.40\pm$0.12 | U/Id
| $0.8$ | $3.2$ | $9.6$ | $10.5$ | $3.0$ | Sige
|  $-0.07\pm$0.15 |  $0.04\pm$0.13 |  $-0.35\pm$0.09 |  $-0.42\pm$0.11 |  $-0.13\pm$0.13 | Q/Ic
-2a |  $-0.33\pm$0.15 |  $-0.01\pm$0.13 |  $0.05\pm$0.09 |  $-0.09\pm$0.12 |  $-0.11\pm$0.13 | U/Id
| $1.4$ | $-$ | $3.1$ | $3.2$ | $0.1$ | Sige

  * a

The row number where 0 presents the center bin containing the pulsar.

  * b

The column number where 0 presents the center bin containing the pulsar.

  * c

The $Q$/$I$ Stokes parameters with 68.3% confidence level error.

  * d

The $U$/$I$ Stokes parameters with 68.3% confidence level error.

  * e

The significance ($\sigma$) for the non-polarized hypothesis test, based on
the standard normal distribution. The corresponding P value means the
probability that a polarized signal is generated by a non-polarized source.
Specifically, a significance of 2$\sigma$ means that the probability that the
emission is polarized reaches 95%. Conventionally, the minimum detectable
polarization at 99% confidence level (MDP99) corresponds to a significance of
2.33$\sigma$.

  * $-$

No significant polarization detection.

Extended Data Table 3: The Polarization degree and angle within the bins of
Fig 1.

| -2b | -1b | 0b | 1b | 2b |
---|---|---|---|---|---|---
2a | 37$\pm$18 | 27$\pm$13 | 61$\pm$12 | 37$\pm$13 | 47$\pm$15 | PDc
$-14\pm$14 |  $-21\pm$14 |  $-41.7\pm$5.3 |  $-52\pm$10 |  $-53.8\pm$8.9 | PAd
1a | 33$\pm$10 | 48.5$\pm$5.0 | 53.5$\pm$4.1 | 56.8$\pm$7.1 | 47$\pm$13 | PDc
6.3$\pm$9.0 |  $-22.4\pm$3.0 |  $-42.2\pm$2.2 |  $-50.2\pm$3.6 |  $-58.2\pm$7.7 | PAd
0a |  $10.3\pm$8.8 | 34.4$\pm$3.9 | 49.0$\pm$2.5 | 62.8$\pm$4.0 | 44$\pm$11 | PDc
$-$7.4$\pm$24 |  $-34.3\pm$3.3 |  $-50.3\pm$1.5 |  $-53.9\pm$1.9 |  $-50.5\pm$7.4 | PAd
-1a | 21$\pm$12 | 27.5$\pm$7.2 | 38.5$\pm$4.0 | 57.1$\pm$5.4 | 44$\pm$12 | PDc
$-47\pm$17 |  $-68.3\pm$7.5 |  $-70.0\pm$3.0 |  $-69.8\pm$2.7 |  $-57.3\pm$7.9 | PAd
-2a | 34$\pm$15 | $4.5^{+13}_{-4.5}$ | 34.9$\pm$9.5 | 43$\pm$12 | 17$\pm$14 | PDc
$-51\pm$13 |  $-6.0\pm$85 |  $86.1\pm$7.8 |  $-84.2\pm$7.6 |  $-70\pm$23 | PAd

  * Polarization degree and polarization angle measured using ixpeobssim. Uncertainties on the polarization degree and angle are calculated for a 68.3% confidence level, assuming that they are independent.

  * a

The row number where 0 presents the center bin containing the pulsar.

  * b

The column number where 0 presents the center bin containing the pulsar.

  * c

The polarization degree (%) with 68.3% confidence level error.

  * d

The polarization angel (∘) with 68.3% confidence level error.

Extended Data Table 4: The polarization degree and angle in the five regions
shown in Extended Data Fig. 5.

| Center | Front | Right | Left | Back
---|---|---|---|---|---
| [ C ] | [ F ] | [ R ] | [ L ] | [ B ]
PD (%)S | 49.6$\pm$2.5 | 70.0$\pm$3.6 | 56.0$\pm$3.1 | 42.0$\pm$3.0 | 33.3$\pm$3.6
PD (%)S+B | 48.8$\pm$2.5 | 65.4$\pm$3.6 | 53.1$\pm$3.1 | 39.9$\pm$3.0 | 31.1$\pm$3.7
PA (∘)S |  $-$50.2$\pm$1.5 |  $-$48.8$\pm$1.5 |  $-$64.9$\pm$1.6 |  $-$30.1$\pm$2.1 |  $-$59.4$\pm$3.1
PA (∘)S+B |  $-$50.3$\pm$1.5 |  $-$48.8$\pm$1.6 |  $-$64.9$\pm$1.7 |  $-$30.2$\pm$2.2 |  $-$59.4$\pm$3.4

  * Polarization degree and polarization angle in C, F, R, L, and B regions measured using ixpeobssim with and without background subtraction. Background regions are the same as shown in Extended Data Fig. 2. Uncertainties on the polarization degree and angle are calculated for a 68.3% confidence level, assuming that they are independent.

  * S

Values are obtained with background subtraction.

  * S+B

Values are obtained without background subtraction.


# A note on a flip-connected class of generalized domino tilings of the box
$[0,2]^{n}$

Andrzej P. Kisielewicz
<EMAIL_ADDRESS>
Wydział Matematyki, Informatyki i Ekonometrii, Uniwersytet Zielonogórski,
ul. Podgórna 50, 65-246 Zielona Góra, Poland

###### Abstract

Let $n,d\in\mathbb{N}$ and $n>d$. An $(n-d)$-domino is a box
$I_{1}\times\cdots\times I_{n}$ such that $I_{j}\in\\{[0,1],[1,2]\\}$ for all
$j\in N\subset[n]$ with $|N|=d$ and $I_{i}=[0,2]$ for every $i\in[n]\setminus
N$. If $A$ and $B$ are two $(n-d)$-dominoes such that $A\cup B$ is an
$(n-(d-1))$-domino, then $A,B$ is called a twin pair. If $C,D$ are two
$(n-d)$-dominoes which form a twin pair such that $A\cup B=C\cup D$ and
$\\{C,D\\}\neq\\{A,B\\}$, then the pair $C,D$ is called a flip of $A,B$. A
family $\mathscr{D}$ of $(n-d)$-dominoes is a tiling of the box $[0,2]^{n}$ if
interiors of every two members of $\mathscr{D}$ are disjoint and
$\bigcup_{B\in\mathscr{D}}B=[0,2]^{n}$. An $(n-d)$-domino tiling
$\mathscr{D}^{\prime}$ is obtained from an $(n-d)$-domino tiling $\mathscr{D}$
by a flip, if there is a twin pair $A,B\in\mathscr{D}$ such that
$\mathscr{D}^{\prime}=(\mathscr{D}\setminus\\{A,B\\})\cup\\{C,D\\}$, where
$C,D$ is a flip of $A,B$. A family of $(n-d)$-domino tilings of the box
$[0,2]^{n}$ is flip-connected, if for every two members
$\mathscr{D},\mathscr{E}$ of this family the tiling $\mathscr{E}$ can be
obtained from $\mathscr{D}$ by a sequence of flips. In the paper some flip-
connected class of $(n-d)$-domino tilings of the box $[0,2]^{n}$ is described.

##  1 Introduction

A non-empty set of the form $K_{1}\times\cdots\times K_{d}\subset[0,2]^{d}$,
where $K_{i}\subset[0,2]$ for $i\in[d]=\\{1,...,d\\}$, is called a box. A
domino is a box $I_{1}\times\cdots\times I_{n}\subset[0,2]^{n}$ such that
$I_{i}=[0,2]$ for some $i\in[n]$ and $I_{j}\in\\{[0,1],[1,2]\\}$ for all
$j\in[n]\setminus\\{i\\}$. A natural generalization of a domino is an
$(n-d)$-domino, where $n>d$, $n,d\in\mathbb{N}$, which is a box
$I_{1}\times\cdots\times I_{n}$ such that $I_{j}\in\\{[0,1],[1,2]\\}$ for all
$j\in N\subset[n]$ with $|N|=d$ and $I_{i}=[0,2]$ for every $i\in[n]\setminus
N$. Thus, a $1$-domino is a domino. Let $A$ and $B$ be two $(n-d)$-dominoes
such that $A\cup B$ is an $(n-(d-1))$-domino. In this case there is $i\in[n]$
such that $A_{j}=B_{j}$ for all $j\in[n]\setminus\\{i\\}$ and
$A_{i},B_{i}\in\\{[0,1],[1,2]\\}$, $A_{i}\neq B_{i}$. Such pair $A,B$ will be
called a twin pair (Figure 1). This $(n-(d-1))$-domino $A\cup B$ can be
divided into two new $(n-d)$-dominoes $C$ and $D$ (Figure 1), that is, $A\cup
B=C\cup D$, $C_{j}=D_{j}$ for all $j\in[n]\setminus\\{k\\}$ and
$C_{k},D_{k}\in\\{[0,1],[1,2]\\}$, $C_{k}\neq D_{k}$, where $k\neq i$. The
twin pair $C,D$ is called a flip of $(n-d)$-dominoes $A$ and $B$. Let us
observe that there are $(n-d)$ flips of $A,B$ (Figure 1).

Figure 1: Two $2$-dominoes $A=[0,2]^{2}\times[0,1]$ and
$B=[0,2]^{2}\times[1,2]$ which form a twin pair (on the left) and two flips of
$A,B$: $C=[0,2]\times[0,1]\times[0,2],D=[0,2]\times[1,2]\times[0,2]$ (in the
middle) and $E=[0,1]\times[0,2]^{2},F=[1,2]\times[0,2]^{2}$ (on the right).

A family $\mathscr{D}$ of $(n-d)$-dominoes is a tiling of the box $[0,2]^{n}$
if interiors of every two members of $\mathscr{D}$ are disjoint and
$\bigcup_{B\in\mathscr{D}}B=[0,2]^{n}$. Clearly, $|\mathscr{D}|=2^{d}$. Let us
suppose that an $(n-d)$-domino tiling $\mathscr{D}$ contains a twin pair
$A,B$. Flipping this pair we obtain a new $(n-d)$-domino tiling
$\mathscr{D}^{\prime}$ of $[0,2]^{n}$, where
$\mathscr{D}^{\prime}=(\mathscr{D}\setminus\\{A,B\\})\cup\\{C,D\\}$ and $C,D$
is a flip of $A,B$. The aim of this note is to present some class of
$(n-d)$-domino tilings of the box $[0,2]^{n}$ which is flip-connected, that
is, for every two tilings $\mathscr{D},\mathscr{E}$ of this class there is a
sequence $\mathscr{D}=\mathscr{T}_{1},...,\mathscr{T}_{m}=\mathscr{E}$ of
$(n-d)$-domino tilings of the class and there are two sequences
$A_{1},B_{1},....,A_{m-1},B_{m-1}$ and $C_{1},D_{1},...,C_{m-1},D_{m-1}$ of
twin pairs, where $A_{i},B_{i}\in\mathscr{T}_{i}$ for $i\in[m-1]$, such that
the pair $C_{i},D_{i}$ is a flip of the pair $A_{i},B_{i}$ for $i\in[m-1]$,
and moreover
$\mathscr{T}_{i+1}=(\mathscr{T}_{i}\setminus\\{A_{i},B_{i}\\})\cup\\{C_{i},D_{i}\\}$
for $i\in[m-1]$.

There are various tilings which can be modify by a sequence of local
transformations. The most known are domino and lozenge tilings ([19, 22]).
Local transformations of domino tilings in dimensions $d\geq 3$ are also
examined ([5, 6, 16, 20, 21]). Generalized domino tilings are considered in
[17].

Our result for $(n-d)$-dominoes is based on special codes which encode some
partitions of a $d$-dimensional box $X$ into smaller boxes ([13, 11]). Let $S$
be a set of arbitrary objects, and let $s\mapsto s^{\prime}$ be a permutation
of $S$ such that $s^{\prime\prime}=(s^{\prime})^{\prime}=s$ and
$s^{\prime}\neq s$. Let $S^{d}=\\{v_{1}...v_{d}\colon v_{i}\in S\\}$. Elements
of $S$ are called letters, while members of $S^{d}$ will be called words. We
add to $S$ an extra letter $*$ and the set $S\cup\\{*\\}$ will be denoted by
$*S$. We assume that $*^{\prime}=*$. (Clearly, elements of $(*S)^{d}$ will be
also called words.) Two words $v,w\in(*S)^{d}$ are dichotomous if there is
$i\in[d]$ such that $v_{i},w_{i}\in S$ and $v_{i}=w^{\prime}_{i}$. If
additionally $v_{j}=w_{j}$ for $j\in[d]\setminus\\{i\\}$, then we say that $v$
and $u$ form a twin pair (in the $i$-th direction). A set $V\subset(*S)^{d}$
is a code if every two distinct words in $V$ are dichotomous (compare Table
1). It is easy to see that $|V|\leq 2^{d}$. If $V\subset S^{d}$ and
$|V|=2^{d}$, then $V$ is called a cube tiling code. If $V$ is a cube tiling
code such that $v_{i}\in\\{w_{i},w_{i}^{\prime}\\}$ for every $v,w\in V$ and
every $i\in[d]$, then $V$ is called simple.

A $d$-box is a set of the form $X=X_{1}\times\cdots\times X_{d}$, where
$|X_{i}|\geq 2$ for $i\in[d]$. A box in a $d$-box $X$ is any non-empty set of
the form $K=K_{1}\times\cdots\times K_{d}\subset X$, where $K_{i}\subset
X_{i}$ for $i\in[d]$. A box $K$ is  proper if $K_{i}\neq X_{i}$ for each
$i\in[d]$. Two boxes $K,G\subset X$ are called dichotomous if there is
$i\in[d]$ such that $K_{i}=X_{i}\setminus G_{i}$. If $\mathscr{F}$ is a family
of pairwise dichotomous proper boxes in $X$ containing $2^{d}$ boxes, then it
is called a minimal partition of $X$ (Figure 2). Each minimal partition of $X$
is a partition of the $d$-box $X$, that is, $\bigcup_{K\in\mathscr{F}}K=X$.

Cube tiling codes encode minimal partitions. To see this, let $\mathscr{P}(A)$
be the power set of a set $A$, and let for each $i\in[d]$ a mapping
$f_{i}\colon S\to\mathscr{P}(X_{i})\setminus\\{\emptyset,X_{i}\\}$ be such
that $f_{i}(s^{\prime})=X_{i}\setminus f_{i}(s)$. We define the mapping
$f\colon S^{d}\to\mathscr{P}(X)$ by $f(s_{1}\ldots
s_{d})=f_{1}(s_{1})\times\cdots\times f_{d}(s_{d}).$ The set
$f(V)=\\{f(v)\colon v\in V\\}$ is said to be a realization of the code $V$
(Figure 2 and Table 1). If $V\subset S^{d}$ is a cube tiling code, then the
set of boxes $f(V)$ is a minimal partition of $X$. Cube tiling codes encode
also other interesting combinatorial structures such as $2$-periodic cube
tilings of $\mathbb{R}^{d}$ or perfect codes in the maximum metric ([3, 8]).

It is easy to check that any minimal partition defines a cube tiling code (see
[9, Subsection 2.6]).

Two dichotomous boxes $K,G\subset X$ are called a twin pair if there is
$i\in[d]$ such that $K_{i}=X_{i}\setminus G_{i}$ and $K_{j}=G_{j}$ for all
$j\in[d]\setminus\\{i\\}$. Let us note that for any twin pair $K,G$, where
$K_{i}=X_{i}\setminus G_{i}$, the set $H=K\cup G$ is a box with $H_{i}=X_{i}$.
The box $H$ is called a gluing of $K,G$. Taking a non-empty set $M_{i}\subset
X_{i}$, where $M_{i}\not\in\\{X_{i},K_{i},G_{i}\\}$ we can divide the box $H$
into two new boxes $M$ and $N$, that is $H=M\cup N$, where $M_{j}=N_{j}=K_{j}$
for $j\in[d]\setminus\\{i\\}$ and $N_{i}=X_{i}\setminus M_{i}$. Let us call
this operation a cutting of $H$. Thus, if $\mathscr{F}$ is a minimal partition
of a $d$-box $X$ and a twin pair $K,G$ belongs to $\mathscr{F}$, then by
gluing and cutting we can pass from $\mathscr{F}$ to a new minimal partition
$\mathscr{G}=(\mathscr{F}\setminus\\{K,G\\})\cup\\{M,N\\}$ ([7]). In Figure 2
we see a passing by gluing and cutting from one minimal partition to another.

Figure 2: A passing by gluing and cutting from one minimal partition of the
box $[0,2]^{2}$ to another. At each step of the process precisely one twin
pair is modified. The partitions on the most left and the most right are a
realization of the codes
$V=\\{aa,aa^{\prime},a^{\prime}b,a^{\prime}b^{\prime}\\}$ and
$W=\\{cc,c^{\prime}c,bc^{\prime},b^{\prime}c^{\prime}\\}$, respectively. In
the realizations of the codes we took
$f_{1}(a)=f_{2}(a)=[0,1],f_{1}(b)=f_{2}(b)=[0,\frac{1}{2}]$ and
$f_{1}(c)=f_{2}(c)=[0,\frac{3}{2}].$

For simplicity we shall work with codes $V\subset S^{d}$ rather than boxes.
Moreover, if $v,w$ is a twin pair with $v_{i}=w_{i}^{\prime}$, then a twin
pair $u,q$ such that $u_{i}=q_{i}^{\prime}$,
$\\{v_{i},w_{i}\\}\neq\\{u_{i},q_{i}\\}$ and $v_{j}=w_{j}=u_{j}=q_{j}$ for
$j\in[d]\setminus\\{i\\}$ will be called a shift of the pair $v,w$ ([15]). If
$u,q$ is a shift of $v,w$, then we will write $\\{v,w\\}\rightarrow\\{u,q\\}$.
Thus, in some cases we may pass from a cube tiling code $V$ to a cube tiling
code $W$ by a sequence of shifts. For example, we may pass from $V$ to $W$
(Figure 2) by shifts:

$\\{aa,aa^{\prime}\\}\rightarrow\\{ac,ac^{\prime}\\},\;\;\\{a^{\prime}b,a^{\prime}b^{\prime}\\}\rightarrow\\{a^{\prime}c,a^{\prime}c^{\prime}\\},\;\;\\{ac^{\prime},a^{\prime}c^{\prime}\\}\rightarrow\\{bc^{\prime},b^{\prime}c^{\prime}\\},\;\;\\{ac,a^{\prime}c\\}\rightarrow\\{cc,c^{\prime}c\\}.$

If $W$ is obtained from $V$ by shifts, then there are two sequences of twin
pairs $(u^{n},q^{n})_{n=1}^{k}$, $(v^{n},w^{n})_{n=1}^{k}$ and there is a
sequence of cube tiling codes $(V^{j})_{j=1}^{k+1}$ such that:

$\bullet$ For every $j\in[k]$ the twin pair $v^{j},w^{j}$ is a shift of the
twin pair $u^{j},q^{j}$.

$\bullet$ $V^{1}=V$,
$V^{j+1}=(V^{j}\setminus\\{u^{j},q^{j}\\})\cup\\{v^{j},w^{j}\\}$, where
$\\{u^{j},q^{j}\\}\subset V^{j}$ for every $j\in[k]$ and $V^{k+1}=W$.

##  2 Flips of regular $(n-d)$-domino tilings

In [7, Theorem 1.5] (see also [4, 15]) we proved the following

###### Theorem 1

Let $d\leq 6$. For every cube tiling codes $V,W\subset S^{d}$ one may pass by
shifts from $V$ to $W$.

We shall adapt this theorem to the case of $(n-d)$-domino tilings of
$[0,2]^{n}$.

###### Remark 1

In [7] we use a slightly different terminology and if it is possible to pass
by shifts from $V$ to $W$, then we say that $V$ and $W$ are strongly
equivalent.

Let $*A=\\{0,1,*\\}$, and let $v\in(*A)^{n}$. A geometric interpretation of
the word $v=v_{1}...v_{d}$ is the box
$\breve{v}=\breve{v}_{1}\times\cdots\times\breve{v}_{n}\subset[0,2]^{n}$,
where $\breve{v}_{i}=[0,1]$ if $v_{i}=0$, $\breve{v}_{i}=[1,2]$ if $v_{i}=1$
and $\breve{v}_{i}=[0,2]$ if $v_{i}=*$. Let
$\operatorname{prop}\,(v)=\\{i\in[n]\colon v_{i}\neq*\\}$ for $v\in(*A)^{n}$.
If $|\operatorname{prop}\,(v)|=d$, then the box $\breve{v}$ is an
$(n-d)$-domino. To make notations shorter, we shall be working mainly with
$(n-d)$-dominoes in the form of words rather than boxes. Thus, a family of
$(n-d)$-dominoes in which every pair of different members have disjoint
interiors will be represented by a (dichotomous) code $D\subset(*A)^{n}$,
where $d=|\operatorname{prop}\,(v)|$ for every $v\in D$ and $0^{\prime}=1$
(compare Table 1).

Let $D$ be an $(n-d)$-domino tiling of the box $[0,2]^{n}$, and let
$M(D)=[d_{ij}]_{i\in[2^{d}],j\in[n]}$ be a matrix in which rows are words from
$D$. We call the tiling $D$ regular (see Example 2 and Table 1) if there is a
partition of the set $[n]$

$[n]=N_{1}\dot{\cup}\cdots\dot{\cup}N_{d}\dot{\cup}N^{*}$ (1)

with the following properties:

($\alpha$) For every $i\in[d]$ the set $N_{i}=\\{j_{1}<...<j_{n_{i}}\\}$ is
non-empty, and the set $K_{i}$ of the rows of the sub-matrix
$[d_{ij}]_{i\in[2^{d}],j\in N_{i}}$ of the matrix $M(D)$ is of the form:

$K_{i}=\\{\delta*\ldots*,\;\;*\delta*\ldots*,...,*...*\delta\colon\delta\in\\{0,1\\}\\}.$
(2)

($\beta$) The set $N^{*}$ consists of positions of $*$-columns in $M(D)$ and
if $N^{*}\neq\emptyset$, then the set $K^{*}$ of the rows of the sub-matrix
$[d_{ij}]_{i\in[2^{d}],j\in N^{*}}$ of the matrix $M(D)$ contains one word
$*...*$ of length $|N^{*}|$.

The enumeration of the sets $N_{1},...,N_{d}$ is such that
$1=i_{1}<...<i_{d}$, where $i_{1},...,i_{d}$ are the first elements in
$N_{1},...,N_{d}$. Let us note that $|\operatorname{prop}\,(u)\cap N_{i}|=1$
for every $u\in D$ and $i\in[d]$, $|K_{i}|\geq 2$ for $i\in[d]$ (if
$|K_{i}|=2$, then $K_{i}=\\{0,1\\}$) and $|K^{*}|\geq 0$.

###### Example 1

Since it can happen that $K_{i}=\\{0,1\\}$ for some $i\in[d]$ and
$K^{*}\neq\emptyset$, it follows that for every $n,d\in\mathbb{N}$, $n>d$,
there is a regular $(n-d)$-domino tiling of $[0,2]^{n}$, as
$D=\\{\varepsilon_{1}...\varepsilon_{d}*^{n-d}\colon(\varepsilon_{1},...,\varepsilon_{d})\in\\{0,1\\}^{d}\\}$
is regular. In this case we have $N_{i}=\\{i\\}$, $K_{i}=\\{0,1\\}$ for
$i\in[d]$ and $N^{*}=\\{d+1,...,n\\}$, $K^{*}=\\{*^{n-d}\\}$. Let us note that
having a regular $((n-1)-(d-1))$-domino tiling $D$ of the box $[0,2]^{n-1}$
one may easily construct a regular $(n-d)$-domino tiling $E$ of the box
$[0,2]^{n}$. To do that, let
$[n-1]=M_{1}\dot{\cup}\cdots\dot{\cup}M_{d-1}\dot{\cup}M^{*}$ be the partition
(1) for the tiling $D$, and let $K_{1},...,K_{d-1}$ be the sets of the form
(2) for $D$. Then taking the partition
$[n]=N_{1}\dot{\cup}\cdots\dot{\cup}N_{d}\dot{\cup}N^{*}$, where
$N_{1}=\\{1\\}$, $N_{i+1}=M_{i}+1$ for $i\in[d-1]$ and $N^{*}=M^{*}+1$
($A+1=\\{a+1\colon a\in A\\}$) we obtain the regular $(n-d)$-domino tiling $E$
of the box $[0,2]^{n}$, where sets (2) are of the form
$G_{1}=\\{0,1\\},G_{i+1}=K_{i}$ for $i\in[d-1]$ and $G^{*}=K^{*}$.

The above constructed tilings have the property that $K_{i}=\\{0,1\\}$ for
some $i\in[d]$. It is easy to check that for $n=2,3$ all $(n-d)$-domino
tilings of $[0,2]^{n}$ have that property. For example, in the regular domino
tiling $D=\\{0*0,1*0,*01,*11\\}$ of $[0,1]^{3}$ we have
$K_{1}=\\{0*,1*,*0,*1\\},K_{2}=\\{0,1\\}$ ($N_{1}=\\{1,2\\},N_{2}=\\{3\\}$).
The first dimension $n\in\mathbb{N}$ in which there is an $(n-d)$-domino
tiling $D$ of $[0,2]^{n}$ such that the matrix $M(D)$ does not contain a
column which contains only $0,1$ is $n=4$ (compare Example 2). For example,
for $n=4$ and $d=3$ the set
$D=\\{*01\varepsilon,1*0\varepsilon,01*\varepsilon\colon\varepsilon\in\\{0,1\\}\\}\cup\\{000*,111*\\}$
is a domino tiling of $[0,2]^{4}$ and there is no column in the matrix $M(D)$
which contains only 0 and 1. Let us note that the tiling $D$ is not regular.

If the matrix $M(D)$ of an $(n-d)$-domino tiling $D$ of $[0,2]^{n}$ contains a
$*$-column, then one can obtain from the tiling $D$ a new $((n-1)-d)$-domino
tiling $E$ of $[0,2]^{n-1}$, where $E=\\{u_{1}...u_{j-1}u_{j+1}...u_{n}\colon
u\in D\\}$ and $j\in N^{*}$ (that is, $u_{j}=*$ for every $u\in D$). Every
$(n-d)$-domino tiling $D$ of $[0,2]^{n}$ such that $M(D)$ contains a
$*$-column will be called $*$-reducible, and otherwise $D$ will be called
$*$-irreducible.

As we have just seen, since we admit that $K^{*}\neq\emptyset$, we can
construct a regular $(n-d)$-domino tiling of $[0,2]^{n}$ for every
$n,d\in\mathbb{N}$, $n>d$. This is no longer true for regular $(n-d)$-domino
tilings of $[0,2]^{n}$ which are $*$-irreducible, that is, $K^{*}=\emptyset$.
In the next section we give a characterization of numbers $n,d\in\mathbb{N}$,
$n>d$, for which regular $*$-irreducible $(n-d)$-domino tilings of the box
$[0,2]^{n}$ exist.

###  2.1 Regular $(n-d)$-domino tilings and cube tiling codes

The structure of sets $K_{1},...,K_{d}$ makes regular $(n-d)$-domino tilings
of the box $[0,2]^{n}$ very close to cube tiling codes $V\subset S^{d}$. In
this section we describe the relationship between these two types of tilings.

Let $S=\\{a_{1},a^{\prime}_{1},...,a_{n},a_{n}^{\prime}\\}$, and let $D$ be a
regular $(n-d)$-domino tiling of the box $[0,2]^{n}$. For every
$u=u_{1}...u_{n}\in D$ we define a word $v(u)=v_{1}...v_{d}\in S^{d}$: For
each $i\in[d]$ let

$v_{i}=\begin{cases}a_{j}&\text{if $j\in N_{i}$ and $u_{j}=0$},\\\
a_{j}^{\prime}&\text{if $j\in N_{i}$ and $u_{j}=1$},\\\ \end{cases}$ (3)

where $N_{i}$, $i\in[d]$, is such in (1). It is easy to show that the set
$V_{D}=\\{v(u)\colon u\in D\\}$ is a cube tiling code.To do that, let $u,q\in
D$ be two distinct words. Then there are $i\in[d]$ and $j\in N_{i}$ such that
$u_{j}=0$ and $q_{j}=1$, and consequently $v_{i}(u)=v_{i}(q)^{\prime}$, that
is, the words $v(u)$ and $v(q)$ are dichotomous, and since $|D|=2^{d}$, we
have $|V_{D}|=2^{d}$. Thus, $V_{D}$ is a cube tiling code. We shall say that
$V_{D}$ is the cube tiling code of  the tiling $D$.

###### Example 2

For the code $D$ with words given in the matrix $M(D)$ on the left in Table 1
we have $N_{1}=\\{1,2\\},N_{2}=\\{3,4\\}$, $N_{3}=\\{5,6\\}$ and
$N^{*}=\\{7\\}$. Moreover, $K_{i}=\\{0*,1*,*0,*1\\}$ for $i\in\\{1,2,3\\}$ and
$K^{*}=\\{*\\}$. If $S=\\{a_{1},a_{1}^{\prime},...,a_{7},a_{7}^{\prime}\\}$,
then $V_{D}$ is of the form given in the middle in Table 1.

$1$ | $2$ | $3$ | 4 | 5 | 6 | 7
---|---|---|---|---|---|---
$0$ | $*$ | $0$ | $*$ | $0$ | $*$ | $*$
$1$ | $*$ | $1$ | $*$ | $1$ | $*$ | $*$
$*$ | $0$ | $0$ | $*$ | $1$ | $*$ | $*$
$*$ | $1$ | $0$ | $*$ | $1$ | $*$ | $*$
$0$ | $*$ | $1$ | $*$ | $*$ | $1$ | $*$
$0$ | $*$ | $1$ | $*$ | $*$ | $0$ | $*$
$1$ | $*$ | $*$ | $0$ | $0$ | $*$ | $*$
$1$ | $*$ | $*$ | $1$ | $0$ | $*$ | $*$
$1$ | $2$ | $3$
---|---|---
$a_{1}$ | $a_{3}$ | $a_{5}$
$a_{1}^{\prime}$ | $a_{3}^{\prime}$ | $a_{5}^{\prime}$
$a_{2}$ | $a_{3}$ | $a_{5}^{\prime}$
$a_{2}^{\prime}$ | $a_{3}$ | $a_{5}^{\prime}$
$a_{1}$ | $a_{3}^{\prime}$ | $a_{6}$
$a_{1}$ | $a_{3}^{\prime}$ | $a_{6}^{\prime}$
$a_{1}^{\prime}$ | $a_{4}$ | $a_{5}$
$a_{1}^{\prime}$ | $a_{4}^{\prime}$ | $a_{5}$

Table 1: A matrix $M(D)$ of a regular $4$-domino tiling $D$ of the box
$[0,2]^{7}$ with $*$-column at position $7$ (on the left), its cube tiling
code $V_{D}$ (in the middle) and a realization of $V_{D}$ (on the right). In
this realization we took
$X=[0,1]^{3},f_{1}(a_{1})=[0,\frac{1}{2}),f_{1}(a_{2})=[0,\frac{1}{4}),f_{2}(a_{3})=[0,\frac{1}{2}),f_{2}(a_{4})=[0,\frac{1}{4})$
and $f_{3}(a_{5})=[0,\frac{1}{2}),f_{3}(a_{6})=[0,\frac{1}{4})$.

It follows from the above that the meta-structure of an $n$-dimensional
regular $(n-d)$-domino tiling $D$ of $[0,2]^{n}$ is exactly like the structure
of the $d$-dimensional cube tiling code $V_{D}\subset S^{d}$. More precisely,
let $\operatorname{prop}\,(D)=\bigcup_{v\in D}\operatorname{prop}\,(v)$, and
let $X_{i}=[0,2]^{n_{i}}$ for $i\in[d]$, where $n_{i}=|N_{i}|$ (the sets
$N_{i}$, $i\in[d]$, are of the form (1)). For simplicity of the notation, we
take $\operatorname{prop}\,(D)=[n]$ (that is $D$ is $*$-irreducible) and for
every $i\in[d]$ the set $N_{i}$ consists of consecutive integers (compare
Table 1). Let us consider the following realization of $V_{D}$: For $v\in
V_{D}$ and $i\in[d]$ if $v_{i}=a_{j}$, where $j\in N_{i}$
($a_{j},a_{j}^{\prime}$ are such as in (3)), then

$f_{i}(a_{j})=[0,2]\times\cdots\times[0,2]\times[0,1)\times[0,2]\times\cdots\times[0,2],$

where $[0,1)$ stands at the $j$-th position ($f_{i}(a_{j})$ consists of
$n_{i}$ factors) and

$f_{i}(a_{j}^{\prime})=[0,2]\times\cdots\times[0,2]\times[1,2]\times[0,2]\times\cdots\times[0,2],$

where $[1,2]$ stands at the $j$-th position ($f_{i}(a_{j}^{\prime})$ consists
of $n_{i}$ factors). Thus, $f_{i}(a_{j}^{\prime})=X_{i}\setminus f_{i}(a_{j})$
for $i\in[d]$ and $j\in N_{i}$. Let
$\mathscr{D}=\\{f_{1}(v_{1})\times\cdots\times f_{d}(v_{d})\colon v\in
V_{D}\\}$. Since $V_{D}$ is a cube tiling code, its realization $\mathscr{D}$
is a minimal partition of the $d$-box $X=X_{1}\times\cdots\times X_{d}$ (here
we ignore the dimensions of $X_{i}$ for $i\in[d]$). On the other hand, to pass
from $\mathscr{D}$ to $D$ we take into account the dimensions of $X_{i}$ for
$i\in[d]$. To do that, let $A\in\\{{\rm Cl}(B)\colon B\in\mathscr{D}\\}$,
where ${\rm Cl}(B)$ is the closure of $B$. Then
$A=A(v)=A_{1}\times\cdots\times A_{d}$, where $v\in V_{D}$ and the sets
$A_{i}=A^{1}_{i}\times\cdots\times A^{n_{i}}_{i}$, $i\in[d]$, are of the form
${\rm Cl}(f_{i}(a_{j}))$ or $f_{i}(a_{j}^{\prime})$. Since $v\in V_{D}$, there
is $u\in D$ such that $v=v(u)$. Then

$\breve{u}=A^{1}_{1}\times\cdots\times A^{n_{1}}_{1}\times
A^{1}_{2}\times\cdots\times A^{n_{2}}_{2}\times\cdots\times
A^{1}_{d}\times\cdots\times A^{n_{d}}_{d}.$

Clearly, every cube tiling code $V\subset S^{d}$ induces a regular
$(n-d)$-domino tiling of $[0,2]^{n}$. Indeed, let $V\subset S^{d}$ be a cube
tiling code, and let $S_{i}\subset S$ be the set of all letters that appear in
words of $V$ at the $i$-th position, that is, $S_{i}=\\{v_{i}\colon v\in
V\\}$. Since $V$ is a cube tiling code, if $l\in S_{i}$, then $l^{\prime}\in
S_{i}$, and therefore $S_{i}$ is of the form
$S_{i}=\\{a^{1}_{i},(a^{1}_{i})^{\prime},...,a^{r_{i}}_{i},(a^{r_{i}}_{i})^{\prime}\\}$
for $i\in[d]$, $r_{i}\geq 1$. Let us recall, that letters of an alphabet $S$
(see Section 1) were arbitrary objects with the property
$s^{\prime\prime}=(s^{\prime})^{\prime}=s$ and $s^{\prime}\neq s$. In
particular, we can take words of the form $*...*\delta*..*$ as letters, where
$\delta\in\\{0,1\\}$ and $(*...*0*..*)^{\prime}=*...*1*..*$ (0,1 stand at the
same position). Therefore, for every $i\in[d]$ we define a new alphabet
$\bar{S}_{i}=\\{\delta*\ldots*,\;\;*\delta*\ldots*,...,*...*\delta\colon\delta\in\\{0,1\\}\\}$,
where $|\bar{S}_{i}|=2r_{i}$ for $i\in[d]$ and the length of each word in
$\bar{S}_{i}$ is $r_{i}$. Thus, $\bar{S}_{i}$ arises from $S_{i}$ by replacing
each pair $a^{j}_{i},(a^{j}_{i})^{\prime}$, $j\in[r_{i}]$, by the pair
$*...*0*..*,*...*1*..*$, where $0$ and $1$ stand at the $j$-th position.

Let now

$\bar{v}_{i}=\begin{cases}*...*0*...*&\text{if $v_{i}=a_{i}^{j}$, where $0$
stands at the $j$-th position, $j\in[r_{i}]$},\\\ *...*1*...*&\text{if
$v_{i}=(a_{i}^{j})^{\prime}$, where $1$ stands at the $j$-th position,
$j\in[r_{i}]$}.\\\ \end{cases}$

It is easy to see that the set
$D_{V}=\\{\bar{v}=\bar{v}_{1}...\bar{v}_{d}\colon v\in V\\}$ is a regular
$(n-d)$-domino tiling of $[0,2]^{n}$ with $K^{*}=\emptyset$ (that is, $D_{V}$
is $*$-irreducible). We shall say that $D_{V}$ is induced by $V$. Let us note
also, that if the code $V$ is written down in a way that $S_{i}\cap
S_{j}=\emptyset$ for $i\neq j,i,j\in[d]$, then $V$ is the cube tiling code of
$D_{V}$. For example, if
$V=\\{aa,aa^{\prime},a^{\prime}b,a^{\prime}b^{\prime}\\}$, then
$S_{1}=\\{a,a^{\prime}\\},S_{2}=\\{a,a^{\prime},b,b^{\prime}\\}$, but $V$ is
not a cube tiling of the tiling $D_{V}$, as
$S=\\{a,a^{\prime},b,b^{\prime}\\}$. Such $V$ has to be rewritten in the form
$V=\\{a_{1}a_{2},a_{1}a_{2}^{\prime},a_{1}^{\prime}a_{3},a_{1}^{\prime}a_{3}^{\prime}\\}\subset\\{a_{1},a_{1}^{\prime},a_{2},a_{2}^{\prime},a_{3},a_{3}^{\prime}\\}^{2}$
to be (formally) a cube tiling code of an $(n-d)$-domino tiling.

At the end of this section we give a characterization of numbers
$n,d\in\mathbb{N},n>d$, for which regular $*$-irreducible $(n-d)$-domino
tilings of the box $[0,2]^{n}$ exist. To do that, we need the following
property of cube tiling codes which is a consequence of [18, Theorem 6].

###### Lemma 1

Let $d\in\mathbb{N}$. For every $n\in\\{d,...,2^{d}-1\\}$ there is a cube
tiling code $V\subset S^{d}$ such that $\sum_{i=1}^{d}l_{i}=n$, where
$l_{i}=|\\{v_{i}\colon v\in V\\}|/2$ for $i\in[d]$.

Proof. Let $W\subset S^{d}$ be a cube tiling code, and let
$m_{i}=|\\{w_{i}\colon w\in W\\}|/2$ for $i\in[d]$. For $l\in S$ and $i\in[d]$
let $W^{i,l}=\\{w\in W\colon w_{i}=l\\}$. If $\sum_{i=1}^{d}m_{i}>d$, then
there are $a,b\in S$ such $\\{a,a^{\prime}\\}\cap\\{b,b^{\prime}\\}=\emptyset$
and the sets $W^{i,a}\cup W^{i,a^{\prime}}$ and $W^{i,b}\cup W^{i,b^{\prime}}$
are non-empty (and clearly disjoint). Note that, since $W$ is a code, for
every $w\in W^{i,a}\cup W^{i,a^{\prime}}$ and $u\in W^{i,b}\cup
W^{i,b^{\prime}}$ there is $j\in[d]\setminus\\{i\\}$ such that
$w_{j}=u_{j}^{\prime}$. Therefore, the set $W^{1}\subset S^{d}$ defined by the
formula

$W^{1}=W\setminus(W^{i,b}\cup
W^{i,b^{\prime}})\cup\\{w_{1}...w_{i-1}aw_{i+1}...w_{d}\colon w\in
W^{i,b}\\}\cup\\{w_{1}...w_{i-1}a^{\prime}w_{i+1}...w_{d}\colon w\in
W^{i,b^{\prime}}\\}$

is a cube tiling code, and moreover
$\sum_{i=1}^{d}m^{1}_{i}=\sum_{i=1}^{d}m_{i}-1$, where
$m^{1}_{i}=|\\{w_{i}\colon w\in W^{1}\\}|/2$. Clearly, in this way one may
show that if $\sum_{i=1}^{d}m_{i}=d+k$ for some $k\in\mathbb{N}$, then for
every $r\in[k]$ there is a cube tiling code $W^{r}$ such that
$\sum_{i=1}^{d}m^{r}_{i}=\sum_{i=1}^{d}m_{i}-r$, where
$m^{r}_{i}=|\\{w_{i}\colon w\in W^{r}\\}|/2$.

Now our result follows from [18, Theorem 6 for $n=2$] which implies that
$\sum_{i=1}^{d}l_{i}\leq 2^{d}-1$ for every cube tiling $V\subset S^{d}$,
where $l_{i}=|\\{v_{i}\colon v\in V\\}|/2$ and there is a cube tiling code
$U\subset S^{d}$ such that $\sum_{i=1}^{d}l_{i}=2^{d}-1$, where
$l_{i}=|\\{u_{i}\colon u\in U\\}|/2$. $\square$

###### Remark 2

In [18, Theorem 6] a cube tiling $\mathscr{T}=[0,1)^{d}\oplus T$ of the
$d$-dimensional torus $\mathbb{T}^{d}_{m}$ is considered. Since $\mathscr{T}$
is a minimal partition, to get the inequality $\sum_{i=1}^{d}l_{i}\leq
2^{d}-1$ for a cube tiling code $V\subset S^{d}$, we just treat the tiling
$\mathscr{T}$ as a realization of $V$ (see also [18, Theorem 7]).

In the last step of the proof of Lemma 1 instead of [18, Theorem 6] we may use
[1, Corollary], however, this requires a transition from a cube tiling code
$V\subset S^{d}$ to the regular $(n-d)$-domino tiling $D_{V}$ of $[0,2]^{n}$.

###### Theorem 2

Let $n,d\in\mathbb{N}$ and $n>d$. There is a regular $*$-irreducible
$(n-d)$-domino tiling of the box $[0,2]^{n}$ if and only if
$n\in\\{d+1,...2^{d}-1\\}$.

Proof. Let $D$ be a regular $*$-irreducible $(n-d)$-domino tiling of the box
$[0,2]^{n}$, and let $V_{D}$ be the cube tiling code of $D$. For $i\in[d]$ let
$l_{i}=|\\{v_{i}\colon v\in V_{D}\\}|/2$. Note that, for every $i\in[d]$ each
word in the set $K_{i}$ contains $l_{i}-1$ stars (compare (2) and Table 1).
Since $D$ is $*$-irreducible, we have $\sum_{i=1}^{d}(l_{i}-1)=n-d$, and thus,
$\sum_{i=1}^{d}l_{i}=n$. From [18, Theorem 6] it follows that $n\leq 2^{d}-1$,
and since $n>d$, we have $n\in\\{d+1,...,2^{d}-1\\}$.

Let now $n\in\\{d+1,...2^{d}-1\\}$. By Lemma 2, there is a cube tiling
$V\subset S^{d}$ such that $\sum_{i=1}^{d}l_{i}=n$, where
$l_{i}=|\\{v_{i}\colon v\in V\\}|/2$ for $i\in[d]$. Therefore, the tiling
$D_{V}$ induced by $V$ is a regular $*$-irreducible $(n-d)$-domino tiling of
the box $[0,2]^{n}$. $\square$

###  2.2 Flip-connected classes for regular $(n-d)$-domino tilings

In the proof of our version of Theorem 1 for regular $(n-d)$-domino tilings we
shall need the following property:

###### Lemma 2

Let $V,W\subset S^{d}$ be two cube tiling codes such that one can pass by
shifts from $V$ to $W$, and let $S_{i}\subset S$ be the set of all letters
that appear in words of the set $V\cup W$ at the $i$-th position. Then it is
possible to pass by shifts $\\{v^{1},w^{1}\\},...,\\{v^{k},w^{k}\\}$ from $V$
to $W$ such that $v^{j}_{i},w^{j}_{i}\in S_{i}$ for every $j\in[k]$.

Proof. Assume that there is a sequence of shifts
$\\{r^{1},p^{1}\\},...,\\{r^{k},p^{k}\\}$ which transforms $V$ into $W$ such
that $r^{t}_{i}=l\not\in S_{i}$ for some $t\in[k]$. As it was noted in Section
1, there is a sequence $\\{u^{1},q^{1}\\},...,\\{u^{k},q^{k}\\}$ of twin pairs
and a sequence $V^{1},...,V^{k+1}$ of cube tiling codes such that
$r^{j},p^{j}$ is a shift of $u^{j},q^{j}$ for $j\in[k]$, and moreover

$V^{1}=V,\;\;\;\;V^{j+1}=(V^{j}\setminus\\{u^{j},q^{j}\\})\cup\\{r^{j},p^{j}\\}\;\;{\rm
and}\;\;V^{k+1}=W,$

where $\\{u^{j},q^{j}\\}\subset V^{j}$ for $j\in[k]$. Let

$(V^{j})^{i,l}=\\{v\in V^{j}\colon v_{i}=l\\}\;\;{\rm
and}\;\;(V^{j})^{i,l^{\prime}}=\\{v\in V^{j}\colon v_{i}=l^{\prime}\\}.$

For $s,s^{\prime}\in S\setminus\\{l,l^{\prime}\\}$ let

$U^{j}=(V^{j}\setminus((V^{j})^{i,l}\cup(V^{j})^{i,l^{\prime}}))\cup(Q^{j})^{i,s}\cup(Q^{j})^{i,s^{\prime}},$

where

$(Q^{j})^{i,s}=\\{v_{1}...v_{i-1}sv_{i+1}...v_{d}\colon
v\in(V^{j})^{i,l}\\}\;\;{\rm
and}\;\;(Q^{j})^{i,s^{\prime}}=\\{v_{1}...v_{i-1}s^{\prime}v_{i+1}...v_{d}\colon
v\in(V^{j})^{i,l^{\prime}}\\}.$

It is easy to check that $U^{j}$ is a cube tiling code for $j\in[k+1]$
(compare [10, Subsection 2.1]). Since $l,l^{\prime}\not\in S_{i}$, we have
$U^{1}=V$ and $U^{k+1}=W$.

Let now $(\bar{u}^{j},\bar{q}^{j})_{j\in[k]}$ and
$(\bar{r}^{j},\bar{p}^{j})_{j\in[k]}$ be two sequences of twin pairs, which
arise from $(u^{j},q^{j})_{j\in[k]}$ and $(r^{j},q^{j})_{j\in[k]}$ by the
substitutions $l\rightarrow s,l^{\prime}\rightarrow s^{\prime}$ at the $i$-th
position in every word of $(u^{j},q^{j})_{j\in[k]}$ and
$(r^{j},q^{j})_{j\in[k]}$. By the definition of $U^{j}$, we have
$\\{\bar{u}^{j},\bar{q}^{j}\\}\subset U^{j}$ and
$U^{j+1}=(U^{j}\setminus\\{\bar{u}^{j},\bar{q}^{j}\\})\cup\\{\bar{r}^{j},\bar{p}^{j}\\}$
for $j\in[k]$.

Since $r^{j},p^{j}$ was a shift of $u^{j},q^{j}$ for $j\in[k]$, it follows
that $\bar{r}^{j},\bar{p}^{j}$ is a shift of $\bar{u}^{j},\bar{q}^{j}$ for
$j\in[k]$. Thus, we may pass from $V$ to $W$ by the sequence of shifts
$\\{\bar{r}^{1},\bar{p}^{1}\\},...,\\{\bar{r}^{k},\bar{p}^{k}\\}$. As there
was no additional assumption on the letter $s\in S$, we may assume that
$s,s^{\prime}\in S_{i}$. $\square$

Let $D$ be an $(n-d)$-domino tiling of $[0,2]^{n}$. A subset $C\subset D$ is
called a simple component of $D$ if
$\operatorname{prop}\,(v)=\operatorname{prop}\,(w)$ for every $v,w\in C$ and
$\operatorname{prop}\,(v)\neq\operatorname{prop}\,(u)$ for every $v\in C$ and
$u\in D\setminus C$ (the tiling in Table 1 has four simple components). If
$\operatorname{prop}\,(v)=\operatorname{prop}\,(w)$ for every $v,w\in D$, then
we say that $D$ is a simple $(n-d)$-domino tiling of $[0,2]^{n}$. It is
obvious that every simple $(n-d)$-domino tiling of $[0,2]^{n}$ is regular.
Moreover, if $W\subset S^{d}$ is the cube tiling code of a regular
$(n-d)$-domino tiling $D$ of $[0,2]^{n}$ and $W$ is simple, then $D$ is
simple.

Theorem 1 for regular $(n-d)$-domino tilings of $[0,2]^{n}$ reads as follows:

###### Theorem 3

Let $d\leq 6$. The family of all regular $(n-d)$-domino tilings of the box
$[0,2]^{n}$ is flip-connected.

Proof. Let $D,E\subset(*A)^{n}$ be two regular $(n-d)$-domino tilings of
$[0,2]^{n}$.

Let $V_{D}\subset S^{d}$ be the cube tiling code of $D$, $C\subset D$ be a
simple component of $D$, and let $V_{C}\subset V_{D}$ be the set of all words
which encode $C$. By Theorem 1 we can pass by shifts from $V_{D}$ to $W$,
where $W$ is the simple cube tiling code such that $V_{C}\subset W$. By Lemma
2 we may do that using the letters from the set $S_{i}\subset S$, $i\in[d]$,
which consists of all letters that appear in words in $V_{D}$ at the $i$-th
position (we have $V_{C}\cup V_{D}=V_{D}$ as $V_{C}\subset V_{D}$). Let
$\\{v^{1},w^{1}\\},...,\\{v^{k},w^{k}\\}$ be a sequence of shifts that
transform $V_{D}$ into $W$.

Let $v,w$ be a twin pair in $V_{D}$ such that $v_{i}=w_{i}^{\prime}$, and let
the twin pair $v^{1},w^{1}$ be the shift of $v,w$, that is,
$v^{1}_{i}=(w^{1}_{i})^{\prime}$,
$\\{v_{i},w_{i}\\}\neq\\{v^{1}_{i},w^{1}_{i}\\}$ and
$v_{j}=w_{j}=v^{1}_{j}=w^{1}_{j}$ for $j\in[d]\setminus\\{i\\}$. We show that
the shift $\\{v,w\\}\rightarrow\\{v^{1},w^{1}\\}$ induces a flip in $D$ which
transform $D$ into a regular $(n-d)$-domino tiling of $[0,2]^{n}$. To do that,
let us recall that $S=\\{a_{1},a_{1}^{\prime},...,a_{n},a_{n}^{\prime}\\}$.
Thus, $v_{i},w_{i}=a_{j},a_{j}^{\prime}$ and
$v^{1}_{i},w^{1}_{i}=a_{s},a_{s}^{\prime}$, where $j\neq s$ and $j\in N_{i}$,
where $N_{i}$ is as in (1) (clearly, $j\in N_{i}$ by the definition of
$N_{i}$). Note now, that, by our assumption stemming from Lemma 2, we have
$s\in N_{i}$. Let $e,f\in D$ be the twin pair such that the words $v,w$ encode
it. We flip $e,f$ into $e^{1},f^{1}$, where $e^{1}_{j}=f^{1}_{j}=*$,
$\\{e^{1}_{s},f^{1}_{s}\\}=\\{0,1\\}$ and $e_{m}=f_{m}=e^{1}_{m}=f^{1}_{m}$
for $m\in[n]\setminus\\{j,s\\}$. Let
$D^{1}=(D\setminus\\{e,f\\})\cup\\{e^{1},f^{1}\\}$. Clearly, if $V_{D}$
contains at least two words with the letter $a_{j}$, then partitions of the
form (1) for $D$ and $D^{1}$ are the same. Thus, $D^{1}$ is regular. Moreover,
$V_{D^{1}}=(V\setminus\\{v,w\\})\cup\\{v^{1},w^{1}\\}$ is the cube tiling code
of $D^{1}$.

Suppose now that $v$ is the only word in $V_{D}$ with $v_{i}=a_{j}$, that is,
the $j$-th column in $M(D)$ contains only one pair $0,1$ ($0$ for $v_{i}$ and
$1$ for $w_{i}=v_{i}^{\prime}$). Let
$[n]=N_{1}\dot{\cup}\cdots\dot{\cup}N_{d}\dot{\cup}N^{*}$ be as in (1) for the
tiling $D$. After gluing the boxes $e,f$ at the $j$-th position that pair
$0,1$ vanishes and the $j$-th column in $M(D^{1})$ consists of stars only.
This means that that the partition
$[n]=M_{1}\dot{\cup}\cdots\dot{\cup}M_{d}\dot{\cup}M^{*}$, where
$M^{*}=N^{*}\cup\\{j\\}$, $M_{m}=N_{m}$ for $m\neq i$ and
$M_{i}=N_{i}\setminus\\{j\\}$, has the properties ($\alpha$) and ($\beta$) for
the tiling $D^{1}$. To see this, observe that $M_{i}\neq\emptyset$, as
$|N_{i}|\geq 2$. Moreover, the set $\bar{K}_{i}$ of the rows of the sub-matrix
$[d_{ij}]_{i\in[2^{d}],j\in M_{i}}$ of the matrix $M(D^{1})$ is of the form
(2): It has two elements less than the set $K_{i}$ for the tiling $D$, and the
lengths of words in $\bar{K}_{i}$ are one less than the lengths of words in
$K_{i}$. The set $M^{*}$ is non-empty and contains one word of length
$|N^{*}|+1$ consisting only of stars.

Therefore, the sequence of shifts $\\{v^{1},w^{1}\\},...,\\{v^{k},w^{k}\\}$
induces the sequence $D^{1},...,D^{k+1}$ of regular $(n-d)$-domino tilings of
the box $[0,2]^{n}$ such that $D^{i+1}$ is obtained from $D^{i}$ by a flip for
$i\in[k]$ and $D^{k+1}$ is the simple $(n-d)$-domino tiling of $[0,2]^{n}$
whose cube tiling code is $W$. Now a procedure of passing from $D$ to $E$ by
flips is clear: We pass by flips from $D$ to $C_{D}$ and $E$ to $C_{E}$, where
$C_{D}$ and $C_{E}$ are simple $(n-d)$-domino tilings of $[0,2]^{n}$, and next
we pass by flips from $C_{D}$ to $C_{E}$ (the fact that such passing is
possible is obvious). $\square$

Let $D,E\subset(*A)^{n}$ be $(n-d)$-domino tilings of $[0,2]^{n}$, and let
$D\approx E$ if and only if one can pass by flips from $D$ to $E$. Clearly,
$\approx$ is an equivalent relation on the family of all $(n-d)$-domino
tilings of $[0,2]^{n}$. Let $[D]_{\approx}$ be the equivalent class of $D$.
Note that, if $D$ is regular, then some elements of $[D]_{\approx}$ need not
to be regular. For example, if in the tiling given in Table 1 we flip the twin
pair $v^{3}=*00*1**$ and $v^{4}=*10*1**$ into $u=**001**$ and $q=**011**$,
then we obtain a tiling which is not regular. (That is why in the proof of
Theorem 3 we needed Lemma 2.) We shall call such non-regular elements of
$[D]_{\approx}$, where $D$ is regular, almost regular.

From Theorem 3 we obtain

###### Corollary 1

Let $d\leq 6$. If $D$ and $E$ are regular $(n-d)$-domino tilings of the box
$[0,2]^{n}$, then $[D]_{\approx}=[E]_{\approx}$. In other words, the family of
all almost regular $(n-d)$-domino tilings of the box $[0,2]^{n}$ is flip-
connected.

$\square$

At the end of the paper we give an example of a class of subsets of the box
$[0,2]^{n}$ with the property that every family of regular $(n-d)$-domino
tilings from that class is flip-connected. To do that, we need the following
result which stems from [7, Theorem 5.1]:

###### Theorem 4

Let $d\leq 5$, $D$ be a regular $(n-d)$-domino tiling of $[0,2]^{n}$ with a
simple component $C\subset D$, and let $Q$ be the simple $(n-d)$-domino tiling
of $[0,2]^{n}$ such that $C\subset Q$. It is possible to pass by flips from
$D$ to $Q$ keeping $C$ fixed.

Proof. Let $V_{D},V_{C}$ and $W$ be as in the proof of Theorem 3. By [7,
Theorem 5.1] there is a sequence $\\{v^{1},w^{1}\\},...,\\{v^{k},w^{k}\\}$ of
shifts that transform $V_{D}$ into $W$ keeping $V_{C}$ fixed. Precisely in the
same manner as in the proof of Theorem 3 we show that the sequence of shifts
$\\{v^{1},w^{1}\\},...,\\{v^{k},w^{k}\\}$ induces the sequence
$D^{1},...,D^{k+1}$ of regular $(n-d)$-domino tilings of the box $[0,2]^{n}$
such that $D^{i+1}$ is obtained from $D^{i}$ by a flip for $i\in[k]$ and
$D^{k+1}$ is the simple $(n-d)$-domino tiling of $[0,2]^{n}$ whose cube tiling
code is $W$. Clearly, $D^{k+1}=Q$ and since $V_{C}$ was unchanged during the
passing from $V_{D}$ to $W$, the set $C\subset Q$ was unchanged during the
passing from $D$ to $Q$. $\square$

Let $C$ be a simple component of a regular $(n-d)$-domino tiling $D$ of
$[0,2]^{n}$, and let $F_{C}=[0,2]^{n}\setminus\bigcup_{v\in C}\breve{v}$. Let
us note, that $F_{C}$ does not have to be a box. In what follows we describe a
family of $(n-d)$-domino tilings of $F_{C}$ which is flip-connected.

###### Corollary 2

Let $d\leq 5$, $C$ be a simple component of a regular $(n-d)$-domino tiling
$D$ of $[0,2]^{n}$, and let and let $F_{C}=[0,2]^{n}\setminus\bigcup_{v\in
C}\breve{v}$. Then the family $\mathscr{F}$ of all $(n-d)$-domino tilings of
the set $F_{C}$ such that for every $F\in\mathscr{F}$ the tiling $F\cup C$ is
regular is flip-connected.

Proof. Let $F,G\in\mathscr{F}$, and let $Q$ be the simple $(n-d)$-tiling of
$[0,2]^{n}$ such that $C\subset Q$. Since $F\cup C$ and $G\cup C$ are regular,
by Theorem 4 there are two sequences $F^{1},...,F^{k}\in\mathscr{F}$ and
$G^{1},...,G^{m}\in\mathscr{F}$ such that $F=F^{1},Q=F^{k}$ and
$G=G^{1},Q=G^{m}$ and $F^{i+1}$ is obtained from $F^{i}$ by a flip for
$i\in[k-1]$, and similarly $G^{i+1}$ is obtained from $G^{i}$ by a flip for
$i\in[m-1]$. Thus, $G$ can be obtained from $F$ by a sequence of flips.
$\square$

###### Remark 3

In the definition of $F_{C}$ we used the tiling $D$ but we can omit it,
leaving only simple component $C$. To do that, let $Q$ be a simple
$(n-d)$-domino tiling of $[0,2]^{n}$, where sets (1) are of the form
$N_{1}=\\{i_{1}\\},...,N_{d}=\\{i_{d}\\}$. For $i\in\\{i_{1},...,i_{d}\\}$ let
$P_{i}$ be a family of twin pairs $A,B\in Q$ such that
$\\{A_{i},B_{i}\\}=\\{[0,1],[1,2]\\}$. Using [12, Theorem 1] it can be shown
that if $C=Q\setminus\bigcup_{i\in M}P_{i}$, where
$M\subset\\{i_{1},...,i_{d}\\}$, then $F_{C}=\bigcup_{v\in P}\breve{v}$, where
$P=\bigcup_{i\in M}P_{i}$.

##  3 Final comments

Since Keller’s cube tiling conjecture in dimension $d=7$ is true ([2, 10]),
every almost regular $(n-7)$-domino tiling of $[0,2]^{n}$ has a twin pair.
However, we do not know whether Theorem 3 for $d=7$ is true. The method
applied in [7] (which is based on computations) seems to be non-adequate for
the case $d=7$ because of a computational complexity. For all dimensions
$d\geq 8$ Theorem 3 is not true ([13, 14]): If $M$ is the twin pair free cube
tiling code in dimension eight ([14]), then the regular $8$-domino tiling
$D_{M}$ of the box $[0,2]^{16}$ whose cube tiling code is $M$ does not contain
a twin pair, and thus, $|[D_{M}]_{\approx}|=1$. (More about cube tiling codes
without twin pairs can be found in [13].) In general, one can find a twin pair
free $(n-d)$-domino tiling of $[0,2]^{n}$ for $d\geq 3$. For example,
$D=\\{*01\varepsilon,1*0\varepsilon,01*\varepsilon\colon\varepsilon\in\\{0,1\\}\\}\cup\\{000*,111*\\}$
is a domino tiling of $[0,2]^{4}$ without twin pairs (that is,
$|[D]_{\approx}|=1$).

The family of all cube tiling codes $V\subset S^{d}$ is known for $d\leq 4$:
In [8] we described the family $\mathscr{N}_{d}$ of all non-isomorphic cube
tiling codes for $d=4$. We obtained $|\mathscr{N}_{4}|=27358$ and the number
of all cube tiling codes for $d=4$ is equal to 17, 794, 836, 080, 455, 680.
For $d=5$ we know that $|\mathscr{N}^{4}_{5}|=899,710,227$, where
$\mathscr{N}^{4}_{5}$ is the family of all non-isomorphic cube tiling codes
$V\subset S^{5}$ with $|S|=4$ ([15]). It is known also ([7]) that the
cardinality of the family of all cube tiling codes $V\subset S^{6}$ far
exceeds the number $10^{84}$. However, we do not know what a portion of all
$(n-d)$-domino tilings of $[0,2]^{n}$ are almost regular tilings.

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