# On the Lucky and Displacement Statistics of
Stirling Permutations

Laura Colmenarejo  Department of Mathematics, North Carolina State University,
Raleigh, NC 27685<EMAIL_ADDRESS>,  Aleyah Dawkins  Department of
Mathematical Sciences, George Mason University, Fairfax, VA 22030
<EMAIL_ADDRESS>,  Jennifer Elder  Department of Computer Science, Mathematics
and Physics, Missouri Western State University, St. Joseph, MO 64507
<EMAIL_ADDRESS>,  Pamela E. Harris  Department of Mathematical
Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI 53211
<EMAIL_ADDRESS>,  Kimberly J. Harry  Department of Mathematical Sciences,
University of Wisconsin-Milwaukee, Milwaukee, WI 53211<EMAIL_ADDRESS>,
Selvi Kara  Department of Mathematics, Bryn Mawr College, Bryn Mawr, PA 19010
<EMAIL_ADDRESS>,  Dorian Smith  School of Mathematics, University of
Minnesota, Minneapolis, MN 55455<EMAIL_ADDRESS>and  Bridget Eileen Tenner
Department of Mathematical Sciences, DePaul University, Chicago, IL 60614
<EMAIL_ADDRESS>
###### Abstract.

Stirling permutations are parking functions, and we investigate two parking
function statistics in the context of these objects: lucky cars and
displacement. Among our results, we consider two extreme cases: extremely
lucky Stirling permutations (those with maximally many lucky cars) and
extremely unlucky Stirling permutations (those with exactly one lucky car). We
show that the number of extremely lucky Stirling permutations of order $n$ is
the Catalan number $C_{n}$, and the number of extremely unlucky Stirling
permutations is $(n-1)!$. We also give some results for luck that lies between
these two extremes. Further, we establish that the displacement of any
Stirling permutation of order $n$ is $n^{2}$, and we prove several results
about displacement composition vectors. We conclude with directions for
further study.

###### Key words and phrases:

parking function, Stirling permutation, lucky statistic, displacement, Catalan
number

###### 1991 Mathematics Subject Classification:

05A05, 05A15

The authors would like to thank the GEMS workshop and AIM for the opportunity
to collaborate.

J. Elder was partially supported by an AWM Mentoring Travel Grant

P. E. Harris was partially supported by a Karen Uhlenbeck EDGE Fellowship.

B. E. Tenner was partially supported by the NSF grant DMS-2054436.

##  1\. Introduction

Throughout the paper, we let $\mathbb{N}\coloneqq\\{0,1,2,3,\ldots\\}$,
$[i,j]\coloneqq\\{i,i+1,\ldots,j-1,j\\}$ for integers $i\leq j$, and
$[n]\coloneqq[1,n]$. A parking function of length $n$ is a tuple
$\alpha=(a_{1},a_{2},\ldots,a_{n})\in[n]^{n}$ whose rearrangement into weakly
increasing order $\alpha^{\uparrow}=(b_{1},b_{2},\ldots,b_{n})$ satisfies
$b_{i}\leq i$ for all $i\in[n]$. For example, $\alpha_{1}=(1,3,1,5,6,3)$ and
$\alpha_{2}=(4,4,1,2,2,3,3,1)$ are parking functions since their
rearrangements are $\alpha^{\uparrow}_{1}=(1,1,3,3,5,6)$ and
$\alpha^{\uparrow}_{2}=(1,1,2,2,3,3,4,4)$, respectively. However,
$\alpha_{3}=(2,2,3,1,6,6)$ is not a parking function since
$\alpha^{\uparrow}_{3}=(1,2,2,3,6,6)$ and $b_{5}=6\nleq 5$. We let
$\operatorname{PF}_{n}$ denote the set of parking functions of length $n$.

We recall another equivalent interpretation of parking functions.

######  Definition 1.1.

Consider a one-way street with exactly $n$ parking spots and a line-up of $n$
cars, each with a preferred spot that we record as a sequence
$\alpha=(a_{1},a_{2},\ldots,a_{n})\in[n]^{n}$. Car $i$ tries to park in its
preferred spot $a_{i}$. If spot $a_{i}$ is available, then car $i$ parks
there; otherwise, car $i$ continues down the street until it finds the first
available spot to park and parks there. If all cars park, $\alpha$ is a
_parking function_.

Throughout the paper, we use the language from the above definition when
referring to parking functions. Parking functions were introduced in [17] by
Konheim and Weiss in their study of hashing functions, and they established
that $|\operatorname{PF}_{n}|=(n+1)^{n-1}$. Since their introduction, much
work has been done to study this family of combinatorial objects, their
statistics, and numerous generalizations [4, 19, 5, 6, 7, 9, 21].

One such work, motivating the current, is that of Gessel and Seo on so-called
“lucky” cars in parking functions [12].

######  Definition 1.2.

In a parking function $\alpha=(a_{1},a_{2},\ldots,a_{n})$, car $i$ is said to
be lucky if it parks in its preferred spot, $a_{i}$. Given
$\alpha\in\operatorname{PF}_{n}$, write $\mathsf{Lucky}(\alpha)$ for the set
of lucky cars in $\alpha$, and write
$\mathsf{lucky}(\alpha)\coloneqq|\mathsf{Lucky}(\alpha)|$ for the number of
lucky cars in $\alpha$.

For example, $\mathsf{Lucky}(\alpha_{1})=\\{1\\}$ for the parking function
$\alpha_{1}=(1,1,\ldots,1)\in[n]^{n}$ satisfies, which is one of the
“unluckiest” parking functions since the only car that parks in its preferred
spot is car $1$. The lucky cars of a different parking function
$\alpha_{2}=(n,1,1,\ldots,1)\in[n]^{n}$ are $1$ and $2$; that is,
$\mathsf{Lucky}(\alpha_{2})=\\{1,2\\}$ making $\alpha_{2}$ luckier than
$\alpha_{1}$. Permutations of $[n]$ are the “luckiest” parking functions, as
every car gets to park in their preference; that is, if $\alpha$ is a
permutation of $[n]$, then $\mathsf{Lucky}(\alpha)=[n]$ and
$\mathsf{lucky}(\alpha)=n$.

Gessel and Seo gave the following generating function for the lucky car
statistic on parking functions [12]:

$\sum_{\alpha\in\operatorname{PF}_{n}}q^{\mathsf{lucky}(\alpha)}=q\prod_{i=1}^{n-1}(i+(n-i+1)q).$
(1)

Their work is related to rooted trees, and the proof relies on ordered set
partitions and generating function techniques. More recently, it was
established by Harris, Kretschmann, and Martínez Mori, that the sequences
$\alpha\in[n]^{n}$ (not necessarily parking functions) with exactly $n-1$ cars
parking in their preferred spot is equal to the total number of comparisons
performed by the Quicksort algorithm given all possible orderings of an array
of size $n$ [15].

We recall another related statistic for parking functions known as
displacement.

######  Definition 1.3.

Let $\alpha=(a_{1},\ldots,a_{n})\in[n]^{n}$ be a parking function. If car $i$
parks in spot $p(i)\in[n]$, then the displacement of car $i$ is defined by
$d(I)=p(i)-a_{i}$, the difference between where it parks and its preference.
The total displacement of $\alpha$ is given by $d(\alpha)=\sum_{i}d(I)$. In
this way, cars with zero displacement are lucky cars, while any car whose
displacement is positive is unlucky.

Displacement has been widely studied in the literature. Knuth established a
bijection between the total displacement of parking functions and the number
of inversions of labeled trees [16]. Beissinger and Peled constructed a
bijection from parking functions to labeled trees that carries the
displacement of parking functions to the external activity of trees [2]. Yan
studied recurrence relations for generalized displacement in $u$-parking
functions [3], and Aguillon, Alvarenga, Harris, Kotapati, Mori, Monroe,
Saylor, Tieu, Williams found a bijection with the set of ideal states in the
Tower of Hanoi game and parking functions of length $n$ with displacement
equal to one [1]. For more on the lucky statistic and displacement, we point
the reader to the 2023 work of Stanley and Yin [26].

In this paper, we focus on a subset of parking functions consisting of
Stirling permutations.

######  Definition 1.4.

A permutation of the multiset $\\{1,1,2,2,3,3,\ldots,n,n\\}$ is a Stirling
permutation of order $n$ if any value $j$ appearing between the two instances
of $i$ satisfies $j>i$. We write $Q_{n}$ to denote the set of Stirling
permutations of order $n$, and we write $w\in Q_{n}$ as a word $w(1)w(2)\cdots
w(2n)$.

For example, $123321\in Q_{3}$, but $123231\notin Q_{3}$ as 2 appears between
the two instances of 3.

Gessel and Stanley studied Stirling permutations in [13]. The Stirling
polynomial $Q_{n}(t)=\sum_{w\in Q_{n}}t^{des(w)}$, where $des(w)$ refers to
the descent statistic in permutations, arises as the numerator for the
generating function

$\sum_{m\geq 0}S(m+n,n)t^{m}=\frac{Q_{n}(t)}{(1-t)^{2n+1}},$

where $S(m+n,n)$ are the Stirling numbers of the second kind [20, A008277].
The Stirling numbers of the second kind $S(n,k)$ count the number of set
partitions of $[n]$ consisting of exactly $k$ blocks [24]. In [13], Gessel and
Stanley showed $|Q_{n}|=(2n-1)!!$. Since then, others have studied Stirling
permutations to include work on enumerating descents and mesa sets, and other
statistics related to generalized permutations [11, 10, 14, 22].

Stirling permutations of order $n$ can be interpreted as parking functions of
length $2n$. Given a Stirling permutation $w=w(1)w(2)\cdots w(2n)\in Q_{n}$,
the weakly increasing rearrangement of $w$ is always
$(1,1,2,2,3,3,\ldots,n,n)$, which satisfies the required inequality condition
characterizing parking functions. Thus, we have $w\in\operatorname{PF}_{2n}$.
Henceforth, we abuse notation and drop parenthesis and commas when talking
about parking functions, and instead adopt the one-line notation used in
Stirling permutations.

Since Stirling permutations can be viewed as a subset of parking functions, it
is natural to explore the behavior of parking function statistics within this
subset. Our focus is on the displacement and lucky statistics of Stirling
permutations. Furthermore, we investigate their admissible sets in the context
of lucky statistics. Given a subset $S\subset[2n]$, we say $S$ is an
_admissible lucky set_ (or just _admissible_) if there exists $w\in Q_{n}$
such that $\mathsf{Lucky}(w)=S$. Otherwise, we say that $S$ is _not
admissible_.

Our paper is organized in the following way. In Section 2, we consider two
extreme cases and prove the following:

  * •

The so-called “extremely unlucky” Stirling permutations (those with exactly
one lucky car) are enumerated by $(n-1)!$ (Theorem 2.9).

  * •

The so-called “extremely lucky” Stirling permutations (those with exactly $n$
lucky cars) are enumerated by Catalan numbers (Theorem 2.14).

In Sections 3 and 4, we look between these extremes by considering possible
admissible sets of Stirling permutations. As a foray into this broad topic, we
establish the following results:

  * •

If $S$ is an admissible lucky set for $Q_{n}$, then it is also admissible for
all $Q_{m}$ with $m>n$. (Theorem 3.3)

  * •

We provide conditions under which there are exactly two lucky cars (Theorem
4.1). The set $\\{1,n-1,2n-2\\}$ is admissible if and only if $n$ is even
(Proposition 4.4).

In Section 5, we consider the displacement statistic of Stirling permutations
and prove the following results:

  * •

The displacement of any Stirling permutation is $n^{2}$ (Corollary 5.3).

  * •

For any Stirling permutation, the number of cars displaced is between
(inclusively) $n$ and $2n-1$ (Corollary 5.6).

  * •

We show that extremely lucky Stirling permutations are uniquely determined by
their displacement composition, a tuple that describes the displacement of the
unlucky cars (Theorem 5.8 and Corollary 5.10).

We conclude the paper in Section 6 with a selection of open problems.

##  2\. Extremely unlucky/lucky Stirling permutations

The $\mathsf{lucky}$ statistic of Stirling permutations in $Q_{n}$ has a well-
defined range, as we show in the upcoming Lemma 2.1. It is natural to look at
the Stirling permutations that achieve the extreme values in this range. The
main results of this section are characterizations and enumerations of those
two families of Stirling permutations – one enumerated by $(n-1)!$, and the
other enumerated by the $n$th Catalan number.

######  Lemma 2.1.

Let $w=w(1)\cdots w(2n)\in Q_{n}$.

  1. (a)

Car 1 is always lucky.

  2. (b)

If car $i$ is the first car with $w(i)=1$, then $i\in\mathsf{Lucky}(w)$.

  3. (c)

$1\leq\mathsf{lucky}(w)\leq n$.

###### Proof.

(a) Car $1$ is always lucky in each $w\in Q_{n}$, as it is the first car to
park when all the spots (including its desired spot) are available. Thus,
$1\in\mathsf{Lucky}(w)$ for every $w\in Q_{n}$.

(b) Let $i\in[n]$ be the smallest $i$ such that $w(i)=1$, i.e. car $i$ is the
first car to prefer spot $1$. Then car $i$ parks in spot $1$. Hence,
$i\in\mathsf{Lucky}(w)$.

(c) It follows from (a) that $1\leq\mathsf{lucky}(w)$ for every $w\in Q_{n}$.
For the upper bound, note that each car prefers one of the first $n$ parking
spots by the definition of Stirling permutations. Thus, at most $n$ cars get
to park in their desired spots, meaning $\mathsf{lucky}(w)\leq n$. ∎

In the context of unlucky cars, the following complementary observation is
immediate yet useful.

######  Remark 2.2.

Any car whose preference is the second instance of a value in $w\in Q_{n}$ is
unlucky. In particular, the last car $2n$ is always unlucky.

The main results of the next two subsections consider the extremes described
by Lemmas 2.1(c). Namely, we establish that the number of “extremely unlucky”
Stirling permutations (those with only one lucky car) is $(n-1)!$ (see Theorem
2.9) and that “extremely lucky” Stirling permutations (those with $n$ lucky
cars) are counted by Catalan numbers (see Corollary 2.16).

###  2.1. Stirling permutations with extremely bad luck

The lower bound in Lemma 2.1(c) suggests a sort of “extremely bad luck” for
Stirling permutations: having exactly one lucky car.

######  Definition 2.3.

A Stirling permutation $w$ with $\mathsf{Lucky}(w)=\\{1\\}$ is an _extremely
unlucky_ (Stirling) permutation.

One necessary condition for having exactly one lucky car is that $w(1)=1$.
Similarly, we must also have $w(2)=1$, since otherwise $w(2)$ would be lucky.

######  Example 2.4.

There are two extremely unlucky Stirling permutations of order $3$: $112233$
and $112332$.

We now prove a technical result that helps us characterize extremely unlucky
Stirling permutations.

######  Lemma 2.5.

Let $w=w(1)w(2)\cdots w(2n)\in Q_{n}$ and $x\in[2n]$. If $w(i)<x$ for all
$i\leq x$, then $x\not\in\mathsf{Lucky}(w)$.

###### Proof.

Assume that $w(i)<x$ for all $i\leq x$, for some $x\geq 2$ (the statement is
meaningless for $x<2$). Thus the first $x$ cars all want to park in the first
$x-1$ spaces. By the pigeonhole principle, at least one car is unlucky and has
to take the next available space. Consequently, one of these initial $x$ cars
parks in spot $x$. By the assumption, the first appearance of $w(j)=x$ occurs
for $j>x$. Thus, car $j$ does not get to park in its preferred spot because
spot $x$ has already been claimed. Hence $x\not\in\mathsf{Lucky}(w)$. ∎

######  Example 2.6.

Let $w=122133\in Q_{3}$. Consider, first, $x=3$, and note that
$\\{w(1),w(2),w(3)\\}=\\{1,2\\}$. Indeed, $3\not\in\mathsf{Lucky}(w)$,
confirming the result of Lemma 2.5. Now consider $x=2$. Given
$\\{w(1),w(2)\\}=\\{1,2\\}$, the assumption of Lemma 2.5 does not hold. This
still leaves open the possibility for car 2 to be unlucky. However, car 2
parks in its preferred spot 2, indicating that it is not unlucky after all.

In other words, if the first $x$ values of $w$ are all less than $x$, then $x$
is not lucky. In fact, as the above example suggests, this is a necessary and
sufficient condition to characterize the extremely unlucky Stirling
permutations.

######  Corollary 2.7.

A Stirling permutation $w\in Q_{n}$ is extremely unlucky if and only if for
all $x\in[2,n]$, we have $w(i)<x$ for all $i\leq x$.

###### Proof.

Let $w\in Q_{n}$ be a Stirling permutation. If for all $x\in[2,n]$, we have
$w(i)<x$ for all $i\leq x$, then $w$ is extremely unlucky by Lemma 2.5.

Now, suppose that $w$ is extremely unlucky. Recall that
$2\not\in\mathsf{Lucky}(w)$ and it means that $w(2)=1$, giving the result for
$x=2$. Fix $x\in[3,n]$ and suppose, inductively, that for all $y\in[2,x-1]$,
we have $w(i)<y$ for all $i\leq y$. Suppose, for the purpose of obtaining a
contradiction, that $w(x)\geq x$. For $w$ to be extremely unlucky, for each
$y\in[2,x-1]$, it must be the case that the first $y$ cars have parked in
spots $[1,y]$. Thus the first $x-1$ cars have parked in spots $[1,x-1]$,
meaning that spot $w(x)$ is available for car $x$ to park in. Thus, car $x$ is
lucky, which contradicts the assumption that $w$ is extremely unlucky. ∎

The following set is useful in our proof of the enumeration of the extremely
unlucky Stirling permutations.

######  Definition 2.8.

For a finite, nonempty set $S$ of integer numbers, we define
$S^{\triangledown}$ to be the subset of $S$ consisting of everything but the
largest element of $S$; that is,

$S^{\triangledown}\coloneqq S\setminus\\{\max(S)\\}.$

Next, we enumerate extremely unlucky Stirling permutations via a constructive
proof in which we build extremely unlucky Stirling permutations of order $n$
by first positioning the two copies of $n$, then the two copies of $n-1$, and
so on, always ensuring that the only lucky car is the first car whose
preferred spot is $1$. The construction is demonstrated after the proof of the
theorem, in Example 2.10.

######  Theorem 2.9.

For any integer $n\geq 1$, there are $(n-1)!$ extremely unlucky Stirling
permutations of order $n$.

###### Proof.

Following Corollary 2.7, we can construct the extremely unlucky Stirling
permutations of order $n$ as follows. First, we position two (necessarily
consecutive) copies of $n$, with the leftmost of those appearing as $w(i_{n})$
for some $i_{n}\in[n+1,2n]^{\triangledown}$. There are $2n-(n+1)+1-1=n-1$ ways
to do this. The rightmost $n$ necessarily appears at $w(i_{n}^{+})$, where
$i_{n}^{+}$ is the smallest element larger than $i_{n}$ in the set $[n+1,2n]$;
that is, $i_{n}^{+}=i_{n}+1$.

Next, we position the two copies of $n-1$. Because we are building a Stirling
permutation, these are either consecutive or they immediately surround the two
copies of $n$. Thus, it follows from Corollary 2.7 that the leftmost of these
appears as $w(i_{n-1})$ for some

$i_{n-1}\in\Big{(}[n,2n]\ \setminus\
\\{i_{n},i_{n}^{+}\\}\Big{)}^{\triangledown},$

and the rightmost $n-1$ necessarily have position $w(i_{n-1}^{+})$, where
$i_{n-1}^{+}$ is the smallest element larger than $i_{n-1}$ in the set
$[n,2n]\setminus\\{i_{n},i_{n}^{+}\\}$. There are $2n-n+1-2-1=n-2$ choices for
$i_{n-1}$.

We continue this process until we find two available positions for the
leftmost $3$, one available position for the leftmost $2$, and finally insert
the two $1$s as $w(1)$ and $w(2)$. For example, the leftmost $x$ appears as
$w(i_{x})$ for some

$i_{x}\in\Big{(}[x+1,2n]\ \setminus\ \\{i_{n},i_{n}^{+}\\}\ \setminus\
\\{i_{n-1},i_{n-1}^{+}\\}\ \setminus\ \cdots\ \setminus\
\\{i_{x+1},i_{x+1}^{+}\\}\Big{)}^{\triangledown},$

and thus there are $2n-(x+1)+1-2(n-x)-1=x-1$ choices for $i_{x}$. The second
appearance of $x$ is at $w(i_{x}^{+})$, where $i_{x}^{+}$ is the smallest
element larger than $i_{x}$ in the set

$[x+1,2n]\ \setminus\ \\{i_{n},i_{n}^{+}\\}\ \setminus\
\\{i_{n-1},i_{n-1}^{+}\\}\ \setminus\ \cdots\ \setminus\
\\{i_{x+1},i_{x+1}^{+}\\}.$

This process has constructed $(n-1)(n-2)\cdots 3\cdot 2\cdot 1=(n-1)!$
distinct Stirling permutations, which are exactly the extremely unlucky
Stirling permutations of order $n$. ∎

We demonstrate the construction in the proof of Theorem 2.9 with an example.

######  Example 2.10.

Let $n=6$.

  * •

The index $i_{6}$ can be any of the five values in
$[7,12]^{\triangledown}=\\{7,8,9,10,11\\}$. Suppose $i_{6}=9$, and so
$i_{6}^{+}=10$.

  * •

The index $i_{5}$ can be any of the four values in
$\big{(}[6,12]\,\setminus\,\\{9,10\\}\big{)}^{\triangledown}=\\{6,7,8,11\\}$.
Suppose $i_{5}=6$, and so $i_{5}^{+}=7$.

  * •

The index $i_{4}$ can be any of the three values in
$\big{(}[5,12]\,\setminus\,\\{9,10\\}\,\setminus\,\\{6,7\\}\big{)}^{\triangledown}=\\{5,8,11\\}$.
Suppose $i_{4}=8$, and so $i_{4}^{+}=11$.

  * •

The index $i_{3}$ can be any of the two values in
$\big{(}[4,12]\,\setminus\,\\{9,10\\}\,\setminus\,\\{6,7\\}\,\setminus\,\\{8,11\\}\big{)}^{\triangledown}=\\{4,5\\}$.
Suppose $i_{3}=5$, and so $i_{3}^{+}=12$.

  * •

The index $i_{2}$ must be the one value in
$\big{(}[3,12]\,\setminus\,\\{9,10\\}\,\setminus\,\\{6,7\\}\,\setminus\,\\{8,11\\}\,\setminus\,\\{5,12\\}\big{)}^{\triangledown}=\\{3\\}$.
Thus we must have $i_{2}=3$ (in fact, $i_{2}$ is always 3), and hence
$i_{2}^{+}=4$.

Thus, we have produced the Stirling permutation

$1\ 1\ 2\ 2\ 3\ 5\ 5\ 4\ 6\ 6\ 4\ 3,$

which is, indeed, extremely unlucky.

###  2.2. Stirling permutations with extremely good luck

In contrast to the extremely unlucky Stirling permutations of Theorem 2.9, we
now consider the very fortunate ones. To begin, note that by Lemma 2.1(c), if
$w\in Q_{n}$, then $\mathsf{Lucky}(w)\subseteq[2n]$ with
$\mathsf{lucky}(w)\leq n$. In other words, at most half of the cars are lucky.

######  Definition 2.11.

A Stirling permutation $w\in Q_{n}$ for which $\mathsf{lucky}(w)=n$ is an
_extremely lucky_ (Stirling) permutation.

Note, in particular, that $w\in Q_{n}$ is extremely lucky if and only if lucky
cars park in each of the first $n$ available parking spots. As we will see,
extremely lucky Stirling permutations are _Catalan objects_ ; that is, they
are enumerated by the Catalan numbers [20, A000108].

######  Definition 2.12.

The _Catalan numbers_ are the sequence $\\{C_{0},C_{1},\ldots\\}$ defined by
$C_{n}=\frac{1}{n+1}\binom{2n}{n}$.

Many families of combinatorial objects are Catalan objects, and we refer the
interested reader to [25] for a comprehensive book on the subject. Our proof
that extremely lucky permutations are another family of Catalan objects is
established by giving a bijection to valid parenthesizations of $n$ pairs of
parentheses; that is, expressions that contain $n$ pairs of parentheses that
are correctly matched. It is well-known that valid parenthesizations are
Catalan objects [23], a result which was proven by Eugène Charles Catalan,
although the sequence was known much earlier by Mingantu, a Mongolian
mathematician [18]. In preparation for that main enumerative result, we first
characterize extremely lucky Stirling permutations.

######  Proposition 2.13.

A Stirling permutation $w\in Q_{n}$ is extremely lucky if and only if the
subword of $w(1)\cdots w(2n)$ consisting of the second appearances of each
number is the word $n\cdots 321$; that is, the second appearances of each
number in $w$ occur in decreasing order.

###### Proof.

Suppose first that there is some $x<n$ for which the second appearance
$x=w(j)$ appears to the left of the second appearance of $x+1$ in $w(1)\cdots
w(2n)$. Since $w$ is a Stirling permutation, we must therefore have

$w=\cdots x\cdots x\cdots(x+1)\cdots(x+1)\cdots.$

But then $x+1$ would not be the parking spot of any lucky car because either
car $j$ would park in spot $x+1$ or some car $h$ with $h<j$ and $w(h)<w(j)$
would have parked in spot $x+1$, before any cars preferring spot $x+1$ have
had a chance to do so. Thus, $w$ is not extremely lucky.

For the other direction of the argument, suppose that the second appearances
of each number in $w$ occur in decreasing order. Then certainly the first
appearance of any $x\in[1,n]$ in $w(1)\cdots w(2n)$ also appears to the left
of the second appearance of any number in $[1,x-1]$. In fact, all the entries
to the left of the first appearance of $x$ are either larger than $x$ or the
first appearance of a number in $[1,x-1]$. Thus, spot $x$ is available for the
first car preferring it for each $x\in[1,n]$. In other words, each car whose
preference is the first appearance of a number in $[1,n]$ is a lucky car, and
hence $\mathsf{lucky}(w)=n$. ∎

Proposition 2.13 characterizes extremely lucky Stirling permutations, and
allows us to give a bijective correspondence between extremely lucky Stirling
permutations of order $n$ and valid parenthesizations of $n$ pairs of
parentheses. For example, the $C_{3}=5$ valid parenthesizations of $3$ pairs
of parentheses are:

$(\ (\ (\ )\ )\ ),\ \ (\ (\ )\ (\ )\ ),\ \ (\ (\ )\ )\ (\ ),\ \ (\ )\ (\ (\ )\
),\ \ \mbox{and}\ \ (\ )\ (\ )\ (\ ).$

######  Theorem 2.14.

Extremely lucky Stirling permutations of order $n$ are in bijection with valid
parenthesizations of $n$ pairs of parentheses.

###### Proof.

One direction of this argument is simple: given an extremely lucky $w\in
Q_{n}$, replace the first occurrence of each $x$ by “(” and the second by “).”
Because $w$ is a Stirling permutation, this produces a valid parenthesization.

Now, given a valid pairing of $n$ pairs of parentheses, replace each “)” by
$n,\ldots,1$ in decreasing order from left to right. For each “)” replaced by
$x$, replace its partner “(” by $x$ too. This construction necessarily
produces a Stirling permutation, and Proposition 2.13 means that we get
exactly those Stirling permutations that are extremely lucky. ∎

We demonstrate the argument in the proof of Theorem 2.14 with the following
example.

######  Example 2.15.

By the bijection described above, the valid parenthesization

$(\ )\ (\ )\ (\ (\ )\ (\ (\ )\ )\ )$

is matched with the extremely lucky Stirling permutation

$6\ 6\ 5\ 5\ 1\ 4\ 4\ 2\ 3\ 3\ 2\ 1\in Q_{6}.$

It is important to note that, as in the proof of Theorem 2.14, any Stirling
permutation can be paired with a valid parenthesization. Thus, many Stirling
permutations could describe the same string of parentheses and among those,
only one is extremely lucky. For instance, the Stirling permutations
$112235546643$ and $665514423321$ both correspond to $(\ )\ (\ )\ (\ (\ )\ (\
(\ )\ )\ )$, but only the latter permutation is extremely lucky (cf. Examples
2.10 and 2.15).

######  Corollary 2.16.

For any positive integer $n$, the number of extremely lucky Stirling
permutations of order $n$ is the Catalan number $C_{n}$.

##  3\. Admissible lucky sets

In this section, we give initial results towards characterizing the subsets of
cars that can be lucky for Stirling permutations. To this end, we recall the
following definition from the introduction.

######  Definition 3.1.

A subset $S\subseteq[2n]$ is an _$n$ -admissible lucky set_ (or just
_admissible_ if context is clear) if there exists $w\in Q_{n}$ such that
$\mathsf{Lucky}(w)=S$. Otherwise, we say that $S$ is _not $n$-admissible_ (or
_not admissible_).

In what follows, we characterize when a set with two elements is an admissible
lucky set and prove several properties of three-element admissible lucky sets.
It remains an open problem to give a complete characterization of all
admissible lucky sets.

######  Example 3.2.

For $n=4$, the set $S=\\{1,3,6\\}$ is a 4-admissible lucky set since
$\mathsf{Lucky}(w)=S$ for $w=33144221\in Q_{4}$. In contrast,
$S^{\prime}=\\{1,4,8\\}$ is not $4$-admissible since there is no $w\in Q_{4}$
such that $\mathsf{Lucky}(w)=S^{\prime}$. Table 1 lists all $4$-admissible
lucky sets.

{1} | {1, 2} | {1, 2, 3} | {1, 2, 3, 4}
---|---|---|---
| {1, 3} | {1, 2, 4} | {1, 2, 3, 5}
| {1, 4} | {1, 2, 5} | {1, 2, 3, 6}
| {1, 5} | {1, 2, 6} | {1, 2, 3, 7}
| {1, 7} | {1, 2, 7} | {1, 2, 4, 5}
|  | {1, 3, 4} | {1, 2, 4, 6}
|  | {1, 3, 5} | {1, 2, 4, 7}
|  | {1, 3, 6} | {1, 2, 5, 6}
|  | {1, 3, 7} | {1, 2, 5, 7}
|  | {1, 5, 6} | {1, 3, 4, 5}
|  | {1, 5, 7} | {1, 3, 4, 6}
|  |  | {1, 3, 4, 7}
|  |  | {1, 3, 5, 6}
|  |  | {1, 3, 5, 7}
Table 1. The thirty-one admissible lucky sets for Stirling permutations of
order $4$, arranged by size.

Determining the properties of admissible lucky sets is the focus of the
remainder of this section. We begin with a stability result, which establishes
that if a set is $n$-admissible, then it is $m$-admissible for all $m\geq n$.

######  Theorem 3.3.

If $S$ is an $n$-admissible lucky set, then the sets $S$ and $S\cup\\{2n+1\\}$
are both $(n+1)$-admissible lucky sets.

###### Proof.

Suppose that $S$ is an $n$-admissible lucky set and that $w\in Q_{n}$
satisfies $\mathsf{Lucky}(w)=S$. Then the Stirling permutation
$w^{\prime}=w(n+1)(n+1)\in Q_{n+1}$, created by appending the value $(n+1)$
twice to the end of $w$, ensures that the last two cars whose preference is
$n+1$ end up parking in spots $2n+1$ and $2n+2$, respectively. Thus,
$\mathsf{Lucky}(w^{\prime})=S$ as well.

Now consider the Stirling permutation $w^{\prime\prime}\in Q_{n+1}$ defined by
$w^{\prime\prime}(i)\coloneqq w(i)+1$ for all $1\leq i\leq 2n$, and
$w^{\prime\prime}(2n+1)=w^{\prime\prime}(2n+2)\coloneqq 1$. By construction,
we have $\mathsf{Lucky}(w^{\prime\prime})=\mathsf{Lucky}(w)\cup\\{2n+1\\}$,
meaning that $S\cup\\{2n+1\\}$ is $(n+1)$-admissible, as desired. ∎

Next, we recall a consequence of Lemma 2.1(a) and Remark 2.2, which involves
elements of admissible lucky sets.

######  Corollary 3.4.

If $S$ is $n$-admissible lucky set, then $1\in S$ and $2n\not\in S$.

In fact, the preferred parking spot of the first lucky car (car 1) determines
the preferences of the initial unlucky cars.

######  Lemma 3.5.

Let $w\in Q_{n}$ with $\mathsf{lucky}(w)\geq 2$, and let $x$ be the second-
smallest element of $\mathsf{Lucky}(w)$. Then the first set of unlucky cars
$2,3,\ldots,x-1$ park, in order, in spots

$w(1)+1,\,w(1)+2,\,\ldots,\,w(1)+x-2.$

###### Proof.

In order for cars $2,3,\ldots,x-1$ to be unlucky, they must each prefer spots
that have already been occupied when it is their turn to park. This implies
that the preference of car $2$ must be $w(2)=w(1)$, ensuring that car $2$
parks in spot $w(1)+1$. As the value $w(1)$ has now appeared twice, the value
$w(3)$ (that is, the preference for car $3$) can only be $w(1)+1$, and thus
car $3$ parks in spot $w(1)+2$. Similarly, regardless of its preference, car
$4$ ends up parking in spot $w(1)+3$. Continuing in this manner, car $y$ parks
in spot $w(1)+y-1$ for all $y\in[2,x-1]$. ∎

######  Remark 3.6.

Under the assumptions of Lemma 3.5, note that $w(i)\geq w(1)$ for
$i\in[2,x-1]$.

The nature of Stirling permutations can impact the parity of certain relevant
values, as we see in the following lemma.

######  Lemma 3.7.

Let $w\in Q_{n}$ and $k\in[1,n]$. Let $j$ be minimal such that $w(j)=k$; that
is, $j$ is the first car to prefer spot $k$ in this parking function. If
$w(i)>w(j)$ for all $i\in[1,j-1]$, then $j$ must be odd.

###### Proof.

Let $w$, $k$, and $j$ be as described in the assumptions of the statement.
Suppose, for the purpose of obtaining a contradiction, that $j$ is even. Then
$j-1$ is odd, so there exist some $h$ and $h^{\prime}$ with
$w(h)=w(h^{\prime})$ and $h<j<h^{\prime}$ as each number appears with 2 copies
in $w$. We have $w(h)>w(j)$ by assumption, which contradicts the fact that $w$
is a Stirling permutation. ∎

In particular, the first car to prefer spot $1$ must be an odd-indexed car. In
the spirit of the previous lemmas, we can deduce the parity of the second
lucky car if it is in the second half of the queue.

######  Proposition 3.8.

Let $w\in Q_{n}$ with $\mathsf{lucky}(w)\geq 2$, and let $x$ be the second-
smallest element of $\mathsf{Lucky}(w)$. If $x>n$, then $x$ is odd.

###### Proof.

Suppose that $x>n$. By Lemma 3.5, after the first $x-1$ cars have parked, the
only unused parking spots are $[1,w(1)-1]\cup[w(1)+x-1,2n]$. Because
$w(1)+x-1>1+n-1=n$, and cars can only prefer the first $n$ spots, the only way
for car $x$ to be lucky would be if it prefers to park in spot $y$ for some
$y<w(1)$. From this, Remark 3.6 and Lemma 3.7 ensure that $x$ must be odd. ∎

Having established several important foundational properties of admissible
lucky sets, we spend the remainder of this section developing some of the more
technical properties of these sets, which becomes relevant in our later
examinations of admissible lucky sets of small cardinality. Many of these
upcoming results focus on the second-smallest element of an admissible lucky
set.

######  Lemma 3.9.

Let $w\in Q_{n}$ with $\mathsf{lucky}(w)\geq 2$, and let $x$ be the second-
smallest element of $\mathsf{Lucky}(w)$. If $w(1)=1$, then $x\leq w(x)\leq n$.

###### Proof.

Suppose that $w(1)=1$ and that car $x$ is the second lucky car. By Lemma 3.5,
cars $2,3,\ldots,x-1$ park to the right of car $1$, in spots $2,3,\ldots,x-1$,
respectively. Thus, if car $x$ is to be lucky, then it must be in the first
half of the queue and prefer one of the spots in the set $\\{x,\dots,n\\}$. In
other words, $x\leq w(x)\leq n$. ∎

When the second-smallest element of $\mathsf{Lucky}(w)$ is sufficiently large,
other lucky cars’ preferences must be suitably small.

######  Lemma 3.10.

Let $w\in Q_{n}$ with $\mathsf{lucky}(w)\geq 3$, and let $x$ be the second-
smallest element of $\mathsf{Lucky}(w)$. If $x\geq n$, then
$w(1)\geq\mathsf{lucky}(w)$. Moreover, the lucky cars park among the first
$w(1)$ spots.

###### Proof.

By Lemma 3.5, for all $i\in[2,x-1]$, car $i$ parks in spot $w(1)+i-1$. Because
$x\geq n$, spots $w(1)$ through $n$ are all occupied before car $x$ parks.
Thus, to ensure there are $\mathsf{lucky}(w)-1$ unused parking spots for the
remaining lucky cars to park in, we must have that
$w(1)-1\geq\mathsf{lucky}(w)-1$; that is, $w(1)\geq\mathsf{lucky}(w)$.
Finally, because spots $w(1)+1$ through $n$ are occupied by unlucky cars, all
lucky cars must park among spots $[1,w(1)]$. ∎

Next, we characterize when the penultimate car in a Stirling permutation is
lucky.

######  Theorem 3.11.

For $w\in Q_{n}$, we have $2n-1\in\mathsf{Lucky}(w)$ if and only if
$w(2n-1)=1$.

###### Proof.

First suppose that $w(2n-1)=1$. If $w(2n)>1$, then the value $1$ would appear
between the two instances of $w(2n)$, meaning that $w$ is not a Stirling
permutation. Thus, $w(2n)=1$, and so car $2n-1$ is the first car to prefer
spot $1$. Therefore, car $2n-1$ parks in spot 1, and hence it is lucky.

Now, suppose that $2n-1\in\mathsf{Lucky}(w)$. Thus, car $2n-1$ is the first
car to prefer $w(2n-1)$, and so $w(2n-1)=w(2n)$. If $w(2n-1)>1$, then consider
the two cars, say cars $i$ and $j$ with $i<j$, with preferred spot
$w(2n-1)-1$. If car $i$ is lucky, then either car $j$ or car $h$ for some
$i<h<j$ parks in spot $w(2n-1)$. On the other hand, if car $i$ is not lucky
then, for some $h\leq i$, car $h$ parks in spot $w(2n-1)$. In either case,
this contradicts the assumption that $2n-1\in\mathsf{Lucky}(w)$. Therefore, we
must have $w(2n-1)=1$. ∎

##  4\. Admissible lucky sets of small cardinality

In this section, we fully characterize and enumerate admissible lucky sets of
size two and provide a first result toward a characterization for admissible
lucky sets of size three.

###  4.1. Admissible lucky sets of size two

As discussed earlier $1\in S$ for any admissible set $S$ (see Corollary 3.4).
In order to characterize admissible sets of size two, one needs to identify
which cars can be the second lucky car.

######  Theorem 4.1.

A set $S=\\{h,i\\}$ with $h<i$ is an $n$-admissible lucky set if and only if
$h=1$ and either

  1. (1)

$i>n$ and $i$ is odd, or

  2. (2)

$i\leq n$.

###### Proof.

By Proposition 3.8, the sets described in the statement are the only possible
lucky sets. For the other direction, we provide Stirling permutations to
demonstrate that all such $\\{1,i\\}$ can be obtained as lucky sets. Let
$w=112233\cdots(n-1)(n-1)nn\in Q_{n}$.

  * •

Suppose $i$ is odd, i.e. $i=2k+1$. Let $\alpha$ be the Stirling permutation
obtained from $w$ by inserting $11$ in between $(k+1)(k+1)$ and $(k+2)(k+2)$,
namely,

$\alpha=2\,2\,3\,3\,\cdots\,(k+1)\,(k+1)\,1\,1\,(k+2)\,(k+2)\,(k+3)\,(k+3)\,\cdots\,n\,n.$

Regardless of whether $i\leq n$ or $i>n$, one can verify that
$\mathsf{Lucky}(\alpha)=\\{1,i\\}$.

  * •

Suppose $i\leq n$ and $i$ is even, i.e. $i=2k$. Let $\alpha^{\prime}$ be the
Stirling permutation obtained from $w$ by inserting $k(2k)(2k)$ in between
$(k-1)(k-1)$ and $(k+1)(k+1)$ and placing the remaining copy of $k$ at the end
after $nn$, namely,

$\alpha^{\prime}=1\,1\,\cdots\,(k-1)\,(k-1)\,k\,(2k)\,(2k)\,(k+1)\,(k+1)\,\cdots\\\
\cdots\,(2k-1)\,(2k-1)(2k+1)\,(2k+1)\,\cdots\,n\,n\,k.$

One can verify that $\mathsf{Lucky}(\alpha^{\prime})=\\{1,i\\}$. ∎

Enumeration of $n$-admissible lucky sets of size two follows from the above
characterization.

######  Corollary 4.2.

For $n\geq 1$, there are
$\displaystyle{n-1+\left\lceil\frac{n-1}{2}\right\rceil}$ $n$-admissible lucky
sets of size two.

Next, we conclude that if $n$ is the second-smallest element in
$\mathsf{Lucky}(w)$ for $w\in Q_{n}$, then $\mathsf{Lucky}(w)=\\{1,n\\}$.

######  Lemma 4.3.

Let $S$ be an $n$-admissible lucky set with second-smallest element $n$.

  1. (1)

If $n$ is even, then $w(1)=1,w(n)=n$, and $|S|=2$.

  2. (2)

If $n$ is odd and $w(1)=1$, then $w(n)=n$ and $|S|=2$.

###### Proof.

Let $w\in Q_{n}$ with $\mathsf{Lucky}(w)=S$ and such that $1$ and $n$ are the
two smallest elements in $S$.

We first show that for $n$ even, $w(1)=1$. Suppose that $n$ is even and
assume, for the purpose of obtaining a contradiction, that $w(1)>1$. Then, by
Lemma 3.5, cars 2 through $n-1$ park in spots $w(1)+1,w(1)+2,\ldots,w(1)+n-2$
where $w(1)+n-2\geq n$. So, lucky car $n$ must prefer a spot in the set
$[1,w(1)-1]$. Moreover, $w(n)$ must be smaller than every element of
$\\{w(1),\ldots,w(n-1)\\}$. By Lemma 3.7, $n$ must be odd. This is a
contradiction. Thus, if $n$ is even, then $w(1)=1$.

Now assume that $w(1)=1$, and make no assumptions on the parity of $n$. By
Lemma 3.5, the first $n-1$ cars park in the first $n-1$ spots, so the only way
car $n$ can be lucky is if it prefers to park in spot $n$; i.e., if $w(n)=n$.
There can be no additional lucky cars because the first $n$ spots are occupied
by the first $n$ cars. Thus, $|S|=2$. ∎

###  4.2. On admissible lucky sets of size three

For $n=4$, Table 1 gives the 11 distinct $4$-admissible lucky sets of
cardinality three. In this section, we characterize certain families of three-
element admissible lucky sets depending on the parity of $n$. It remains an
open problem to characterize all three-element admissible lucky sets.

######  Proposition 4.4.

The set $\\{1,n-1,2n-2\\}$ is an $n$-admissible lucky set if and only if $n$
is even.

###### Proof.

First suppose that $w\in Q_{n}$ with $\mathsf{Lucky}(w)=\\{1,n-1,2n-2\\}$.
Assume $n>3$, with the result being easy to check in small cases.

Recall that the first car to prefer spot $1$ is always lucky, and by Lemma
3.7, it must be an odd-indexed car. Since $2n-2$ is always even, either car
$1$ or car $n-1$, with $n$ even, must be the first car to prefer spot $1$.

If $w(1)=1$, by Lemma 3.5, the only possible spots for the lucky car $n-1$ to
prefer are $n-1$ or $n$. Since car $n$ is unlucky, it must have the same
preference as a previous car, or its preferred spot is occupied. In order to
satisfy the Stirling condition, we have $w(n-1)=w(n)$. If $w(n-1)=n-1$, then
the first $n$ cars park in the first $n$ spots, meaning that car $2n-2$ cannot
possibly be lucky. On the other hand, if $w(n-1)=n$, then the only available
spot for lucky car $2n-2$ to park is spot $n-1$. However, the preference of
car $n+1$ is at most $n-1$. So, it ends up parking in the next available spot
$n-1$, preventing car $2n-2$ from being lucky. Thus, $w(1)=1$ is not possible,
and car $n-1$, with $n$ even, is the first car to prefer spot $1$. In other
words, $\\{1,n-1,2n-2\\}$ is $n$-admissible implies $n$ is even.

To prove the other direction of the result, it suffices to produce a Stirling
permutation $w\in Q_{2k}$ for which $\mathsf{Lucky}(w)=\\{1,2k-1,4k-2\\}$.
Such a permutation is given below:

$3\,3\,4\,4\,\cdots\,k\,k\,(k+1)\,(k+1)\,1\,(k+2)\,(k+2)\,\cdots\,(2k)\,(2k)\,2\,2\,1\in
Q_{2k}.\qed$

##  5\. Displacement statistic

In this section, we study the displacement statistics of Stirling
permutations. As discussed in the introduction, this statistic measures how
far cars park from their preferred parking spots in a parking function.

######  Definition 5.1.

For any $w\in Q_{n}$, if car $i$ parks in spot $p(i)\in[2n]$, then the
displacement of car $i$ is $d(i)\coloneqq p(i)-w(i)$. The (total) displacement
of $w\in Q_{n}$ is

$\displaystyle{d(w)=\sum_{i=1}^{2n}d(i)}.$

Because every Stirling permutation is a parking function, the displacement of
a car is always nonnegative. Moreover, by definition, cars with displacement
equal to zero are lucky cars, while cars whose displacement is positive are
unlucky.

We begin by recalling a result from [8] that for any $w\in Q_{n}$, the
displacement of $w$ is invariant under permutations of the entries of $w$,
even though the resulting strings are not necessarily Stirling permutations.
This result is a special case of [8, Lemma 3.1], hence we omit its proof but
state it formally below.

######  Lemma 5.2.

[8, Lemma 3.1] For any $w\in Q_{n}$, the displacement $d(w)$ is invariant
under permuting the numbers in the one-line notation of $w$. In other words,
for any permutation $\sigma$ of $[1,2n]$ and any Stirling permutation $w\in
Q_{n}$, let $\sigma(w)=w(\sigma(1))w(\sigma(2))\cdots w(\sigma(2n))$. Then
$d(\sigma(w))=d(w)$.

We now determine the displacement of Stirling permutations.

######  Corollary 5.3.

For $w\in Q_{n}$, $d(w)=n^{2}$.

###### Proof.

By Lemma 5.2, it suffices to compute $d(w)$ for the Stirling permutation
$w=112233\cdots nn$. For this $w$, car $i$ parks in spot $i$, for all
$i\in[2n]$. Thus, $d(w)=0+1+1+2+2+\cdots+(n-1)+(n-1)+n=n^{2}$, as desired. ∎

Definition 5.1 provides two versions of displacement: one for each car and one
for the entire Stirling permutation. Corollary 5.3 addresses the latter of
these for Stirling permutations. Now we turn our attention to the former and
investigate “compositions” of $n^{2}$ that record the displacement data for
each car when $w$ is in $Q_{n}$.

######  Definition 5.4.

For $w\in Q_{n}$, we define the displacement composition of $w$, denoted by
$\mathsf{dis}(w)$, as the $2n$-tuple encoding the displacements of each car:

$\mathsf{dis}(w)=(d(1),d(2),\ldots,d(2n))\in\mathbb{N}^{2n}.$

As discussed before, $d(i)=0$ if and only if car $i$ is lucky. In particular,
the first entry of $\mathsf{dis}(w)$ is always $0$.

######  Example 5.5.

For $w=11244233$, $\mathsf{dis}(w)=(0,1,1,0,1,4,4,5)$.

Recall that $1\leq\mathsf{lucky}(w)\leq n$ by Lemma 2.1(c). Thus, we can bound
the number of nonzero entries in $\mathsf{dis}(w)$ for any $w\in Q_{n}$.

######  Corollary 5.6.

For any $w\in Q_{n}$, we have that $1\leq|\\{i\in[2n]:d(i)=0\\}|\leq n$.

Two distinct Stirling permutations may share the same displacement
composition, indicating that displacement compositions do not uniquely
identify a Stirling permutation. For instance, the displacement composition
$(0,1,0,1,3,4)$ corresponds to both $113322$ and $331122$. However, extremely
lucky Stirling permutations are uniquely determined by their displacement
compositions. That is, no two distinct extremely lucky Stirling permutations
have the same displacement composition. This is the main result in this
section, and we start with a technical lemma used in the proof of that result.

######  Lemma 5.7.

Let $w\in Q_{n}$ be an extremely lucky permutation. For $i\in[n]$:

  1. (a)

The $i$th unlucky car of $w$ prefers spot $n-i+1.$

  2. (b)

The $i$th unlucky car of $w$ has displacement $2i-1.$

###### Proof.

Let $w\in Q_{n}$ be an extremely lucky permutation. Hence,
$\mathsf{lucky}(w)=n$ and there are $n$ unlucky cars.

  1. (a)

This is an immediate consequence of Proposition 2.13.

  2. (b)

We proceed by induction. Suppose that car $j$ is the first unlucky car. It was
established in (a) that car $j$ must prefer spot $n$. Since car $j$ is the
second car to prefer spot $n$, and no other cars have yet been unlucky, then
car $j$ parks in spot $n+1$. Thus, its displacement is $(n+1)-n=2(1)-1$, as
claimed.

Now fix $i\geq 1$ and assume that the result holds for all $j\leq i$. Thus,
the $j$th unlucky car parks in spot $(n-j+1)+(2j-1)=n+j$ for all $j\leq i$.
Let car $k$ be the $(i+1)$th unlucky car. By (a), this car $k$ must prefer
spot $n-i$. Since car $k$ is unlucky, and by Proposition 2.13, we know that
spots in $[n-i,n+i]$ are already occupied and the next available spot is
$n+i+1$. Thus, car $k$ parks in spot $n+i+1$. Hence, its displacement is
$(n+i+1)-(n-i)=2i+1=2(i+1)-1$, completing the proof. ∎

We are now ready to establish the injectivity of the map
$w\mapsto\mathsf{dis}(w)$ where $w$ belongs to the set of extremely lucky
Stirling permutations of order $n$.

######  Theorem 5.8.

Let $w$ be an extremely lucky Stirling permutation. The displacement
composition of $w$, $\mathsf{dis}(w)$, uniquely determines $w$.

###### Proof.

Suppose that $w\in Q_{n}$ is an extremely lucky Stirling permutation, and
consider its displacement composition $\mathsf{dis}(w)$. Suppose that there
exists another $w^{\prime}\in Q_{n}$ such that
$\mathsf{dis}(w^{\prime})=\mathsf{dis}(w)$.

The $0$ entries in a displacement composition correspond to the lucky cars in
the parking function, so we must have
$\mathsf{Lucky}(w)=\mathsf{Lucky}(w^{\prime})$, and hence $w^{\prime}$ must be
extremely lucky as well. By Proposition 2.13, the values of $w^{\prime}$ in
each of the positions of nonzero entries of $\mathsf{dis}(w)$ must be
$n,n-1,\ldots,1$, in order from left to right. Moreover, these must be the
second appearances of each of these numbers in the one-line notation of
$w^{\prime}$.

Now, to obey the Stirling property, the first occurrence of $n$ in
$w^{\prime}$ must be in the rightmost unclaimed (lucky) position appearing to
the left of the (already determined) second appearance of $n$ in $w^{\prime}$.
Similarly, the first occurrence of $n-1$ in $w^{\prime}$ must be in the
rightmost unclaimed position appearing to the left of the second appearance of
$n-1$, and so on.

Each of these positions is entirely determined by $\mathsf{dis}(w)$, and so in
fact we must have $w=w^{\prime}$. ∎

We demonstrate the result above in the following example.

######  Example 5.9.

The displacement composition $(0,0,0,1,3,0,0,5,7,9)$ for $w\in Q_{5}$ tells us
that $\mathsf{Lucky}(w)=\\{1,2,3,6,7\\}$. Moreover, Proposition 2.13 tells us
that

$w=\underline{\ \phantom{1}\ }\ \underline{\ \phantom{1}\ }\ \underline{\
\phantom{1}\ }\ \ 5\ \ 4\ \ \underline{\ \phantom{1}\ }\ \underline{\
\phantom{1}\ }\ \ 3\ \ 2\ \ 1.$

The Stirling condition tells us that the first appearance of $5$ in $w$ must
be in position $3$. Then we see that the first appearance of $4$ in $w$ must
be in position $2$. Similarly, the first position of $3$ in $w$ must be
$w(7)$, the first position of $2$ must be $w(6)$, and the first position of
$1$ must be $w(1)$. Therefore,

$w=\underline{\ 1\ }\ \underline{\ 4\ }\ \underline{\ 5\ }\ \ 5\ \ 4\ \
\underline{\ 2\ }\ \underline{\ 3\ }\ \ 3\ \ 2\ \ 1.$

To conclude this section, we give a restatement of Corollary 2.16 in terms of
displacement compositions.

######  Corollary 5.10.

The number of Stirling permutations whose displacement composition has exactly
$n$ nonzero parts is $C_{n}$, the $n$th Catalan number.

##  6\. Open Problems

Finally, we propose a selection of open problems related to lucky cars and the
displacement statistic on Stirling permutations.

To begin with, recall that in Section 4, we gave several properties of three-
element admissible lucky sets. The following problems remain unanswered.

######  Problem 1.

Characterize and enumerate the three-element $n$-admissible lucky sets.

More generally, for fixed $n$, how many $n$-admissible lucky sets are there?

Equation (1) gives a generating function for the lucky statistic on parking
functions. Table 2 provides analogous data for the generating function defined
by the lucky statistic on Stirling permutations defined by

$T_{n}(q)=\sum_{w\in Q_{n}}q^{\mathsf{lucky}(w)}.$

Corollary 2.16 implies that, for all $n\geq 1$, the leading coefficient of
$T_{n}(q)$ is the Catalan number $C_{n}$. Theorem 2.9 shows that the
coefficient of $q$ in $T_{n}(q)$ is given by $(n-1)!$. We thus pose the
following questions.

######  Problem 2.

Determine a formula for $T_{n}(q)$ for any $n\geq 1$. Are the polynomials
$T_{n}(q)$ unimodal? Are they real rooted?

$n$ | $T_{n}(q)$
---|---
2 | $2q^{2}+q$
3 | $5q^{3}+8q^{2}+2q$
4 | $14q^{4}+49q^{3}+36q^{2}+6q$
5 | $42q^{5}+268q^{4}+417q^{3}+194q^{2}+24q$
6 | $132q^{6}+1374q^{5}+3876q^{4}+3665q^{3}+1228q^{2}+120q$
7 | $429q^{7}+6752q^{6}+31231q^{5}+52353q^{4}+34675q^{3}+8975q^{2}+720q$
8 | $1430q^{8}+32197q^{7}+227207q^{6}+620357q^{5}+710425q^{4}+355945q^{3}+74424q^{2}+5040q$
Table 2. The polynomials $T_{n}(q)$ for $2\leq n\leq 8$.

The bound determined in Corollary 5.6, suggests the following avenue of
research.

######  Problem 3.

Determine if, for $i\in[n,2n-1]$, there exists $w\in Q_{n}$ such that
$\mathsf{dis}(w)$ has exactly $i$ nonzero entries. Equivalently, can we always
find a Stirling permutation with $i$ unlucky cars?

Table 3 provides some data in the case where $n=3$, listing all possible
displacement compositions and the Stirling permutations that give rise to
them.

It is shown in Theorem 2.9 that there are $(n-1)!$ Stirling permutations in
$Q_{n}$ whose displacement composition has exactly $1$ zero entry. Moreover,
by Corollary 5.10, there are $C_{n}$ Stirling permutations in $Q_{n}$ whose
displacement composition has exactly $n$ nonzero parts. As a follow-up
question to Problem 3, we can thus try to interpolate between these two
extreme cases, enumerating the Stirling permutations whose displacement
compositions have $k$ zero parts.

######  Problem 4.

For $k\in[2,n-1]$, determine

$\big{|}\\{w\in Q_{n}:\mathsf{dis}(w)\text{ has $k$ zero parts}\\}\big{|}.$

Equivalently, how many $w\in Q_{n}$ have exactly $k$ lucky cars?

It is also of interest to understand displacement compositions arising from
general Stirling permutations.

######  Problem 5.

Let $\mathcal{D}_{n}\coloneqq\\{\mathsf{dis}(w):w\in Q_{n}\\}$. Characterize
and enumerate the elements of $\mathcal{D}_{n}$.

On a more specific level, one can try to understand the fibers of the map
$w\mapsto\mathsf{dis}(w)$. In that direction, a related problem is to
determine the number of Stirling permutations with a fixed displacement
composition.

######  Problem 6.

Fix $\textbf{m}\in\mathcal{D}_{n}$. Characterize and enumerate the elements of
the set

$\mathcal{S}_{n}({\textbf{m}})\coloneqq\\{w\in
Q_{n}:\mathsf{dis}(w)=\textbf{m}\\}.$

Put another way, what properties define the collection of Stirling
permutations whose displacement compositions are equal to a given vector?
Also, are there operations on Stirling permutations whose orbits define the
fibers of that map?

|  $\mathsf{dis}(w)$, where $w\in Q_{3}$
---|---
$k$ | has $k$ nonzero parts
$3$ | $\mathsf{dis}(123321)=(0,0,0,1,3,5)$
$\mathsf{dis}(133221)=(0,0,1,0,3,5)$
$\mathsf{dis}(233211)=(0,0,1,3,0,5)$
$\mathsf{dis}(331221)=(0,1,0,0,3,5)$
$\mathsf{dis}(332211)=(0,1,0,3,0,5)$
$4$ | $\mathsf{dis}(122331)=(0,0,1,1,2,5)$
$\mathsf{dis}(122133)=(0,0,1,3,2,3)$
$\mathsf{dis}(133122)=(0,0,1,1,3,4)$
$\mathsf{dis}(221331)=(0,1,0,1,2,5)$
$\mathsf{dis}(113322)=(0,1,0,1,3,4)$
$\mathsf{dis}(331122)=(0,1,0,1,3,4)$
$\mathsf{dis}(221133)=(0,1,0,3,2,3)$
$\mathsf{dis}(223311)=(0,1,1,2,0,5)$
$5$ | $\mathsf{dis}(112233)=(0,1,1,2,2,3)$
$\mathsf{dis}(112332)=(0,1,1,1,2,4)$
Table 3. Displacement compositions arising from Stirling permutations in
$Q_{3}$, aggregated by the number of nonzero parts; i.e., by the number of
unlucky cars.

## References

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[a]B.F.L. Ward

# Role of IR-Improvement in LHC/FCC Physics

B. Shakerin  S. A. Yost

###### Abstract

One may use amplitude-based resummation in QED $\otimes$ QCD to achieve IR-
improvement of unintegrable singularities in the infrared regime to arbitrary
precision in principle. We illustrate such improvement in specific examples in
precision LHC/FCC physics.

BU-HEPP-20-05, Dec., 2020

In the context of QCD, in addition to the original discussion 111F. Berends,
private communication, at 1988 ICHEP Conference Dinner, Munich, Germany. of
whether the naive Jackson-Scharre [1] or the exact YFS-style [2] resummation
is more accurate for a given level of exactness, we have the issue of a hard
cut-off for the IR versus resummed IR integrability. Somewhat unexpectedly, in
the current era of QCD with precision tags $\lesssim 1.0\%$ with accompanying
EW precision tags at the per mille level for processes such as single heavy
gauge boson production at the LHC, the “new” issue of ISR radiation from
quarks has arisen featuring the choice between QED PDF’s with massless light
quarks vs exact Feynman diagrams with short-distance quark masses and non-QED
PDF’s. The precision data should be able to settle these additional issues as
it resolves the original discussion for the QCD case. Even higher precision
may open further issues – such is the nature of progress in precision
applications of quantum field theory.

We have pursued exact amplitude-based resummation realized on an event-by-
event basis via shower/matrix element(ME) matched MC’s in order to achieve
enhanced precision for a given level of exactness while addressing the present
paradigm in precision physics at the LHC and the futuristic FCC. Currently, we
have a realization of IR-improved parton showers in the Herwig6.5 [3]
environment in the MC Herwiri1.031 [4] by two of us (BFLW and SAY). Via the
MC@NLO [5] and the MG5_aMC@NLO [6] frameworks this MC is elevated to the exact
NLO shower/ME matched level as<EMAIL_ADDRESS>[7] and
<EMAIL_ADDRESS>[8], respectively. We have the realization of IR-
improved (IRI) Pythia8 [10] by one of us (BFLW) in the Pythia8 [9]
environment, with its corresponding NLO shower/ME matched MG5_aMC@NLO/IRI-
Pythia8. More recently, two of us (BFLW and SAY), using the Herwig6.5
environment, have realized in the MC ${\cal KK}$MC-hh [11] exact ${\cal
O}(\alpha^{2}L)$ CEEX EW corrections in a hadronic MC.

As two of us (BFLW, SAY) have shown [4, 7], IR-improvement in Herwig6.5 via
Herwiri1.031 leads to improved precision in both the central
$|\eta_{\ell}|\lesssim 2.5$ region for the ATLAS, CMS, D0 and CDF data and in
the more forward region of LHCb where $2.0<\eta_{\ell}<4.5$. Here,
$|\eta_{\ell}|$ is the lepton pseudorapidity in respective single
$Z/\gamma^{*}$ production production with decay to lepton pairs. One of us
(BFLW) [12] has shown the availability of the IR-improved semi-analytical
paradigm for the latter processes. In what follows, we present an update the
application of our methods in the analysis of LHC W+ n jets data, in the FCC
discovery physics and in the interplay of IR-improved parton showers with
exact ${\cal O}(\alpha^{2}L)$ CEEX EW corrections in ${\cal KK}$MC-hh.

We note that 2017 was the 50th anniversary of the seminal paper by S. Weinberg
[13] in which he formulated his foundational model of leptons in creating the
spontaneously broken $SU_{2L}\times U_{1}$ EW theory [14, 15], one of the key
components of the SM, which we may now call the Standard Theory (ST) 222See
D.J. Gross, talk, SM@50 Symposium, Cleveland, OH, June, 2018.. Progress on
precision theory has been essential to the establishment of the ST[13, 14, 15,
16, 17]. As we celebrate 50 years of the ST (SM) [18], we are also obliged to
look to the future with the FCC [19], CLIC, ILC, or CEPC on the horizion [20],
where, for example, we note that the FCC

Figure 1: FCC device planned for the future at CERN as depicted in Ref. [19].

will feature a 100 TeV hadron collider and a tera-Z $e^{+}e^{-}$ colliding
beam device, as illustrated in Fig. 1. The success of the latter devices will
be strongly correlated with the progress of precision theory.

The paper is organized as follows. We briefly review the parton shower
implementation of exact amplitude-based resummation theory, after which we
turn to the interplay of IR-improved DGLAP-CS QCD theory and shower/ME matched
precision via comparisons with LHC data on $W+n$ jets and via predictions for
FCC discovery physics. Finally, we discuss the interplay of IR-improved DGLAP-
CS QCD theory and exact ${\cal O}(\alpha^{2}L)$ CEEX EW corrections in single
$Z/\gamma^{*}$ production at the LHC.

The parton shower implementation of exact amplitude-based resummation theory
has as its starting point the master formula

$\displaystyle d\bar{\sigma}_{\rm res}=e^{\rm
SUM_{IR}(QCED)}\Large{\sum}_{{n,m}=0}^{\infty}\frac{1}{n!m!}\int\prod_{j_{1}=1}^{n}\frac{d^{3}k_{j_{1}}}{k_{j_{1}}}$
(1)
$\displaystyle\prod_{j_{2}=1}^{m}\frac{d^{3}{k^{\prime}}_{j_{2}}}{{k^{\prime}}_{j_{2}}}\int\frac{d^{4}y}{(2\pi)^{4}}e^{iy\cdot(p_{1}+q_{1}-p_{2}-q_{2}-\sum
k_{j_{1}}-\sum{k^{\prime}}_{j_{2}})+D_{\rm QCED}}$ (2)
$\displaystyle{\tilde{\bar{\beta}}_{n,m}(k_{1},\ldots,k_{n};k^{\prime}_{1},\ldots,k^{\prime}_{m})}\frac{d^{3}p_{2}}{p_{2}^{\,0}}\frac{d^{3}q_{2}}{q_{2}^{\,0}},$
(3)

where new (YFS-style) non-Abelian residuals
$\tilde{\bar{\beta}}_{n,m}(k_{1},\ldots,k_{n};k^{\prime}_{1},\ldots,k^{\prime}_{m})$
have $n$ hard gluons and $m$ hard photons. Definitions of the infrared
functions ${\rm SUM_{IR}(QCED)}$ and ${D_{\rm QCED}}$ and of the residuals are
given in Ref. [7]. In the framework of shower/ME matching, we have the
replacements
$\tilde{\bar{\beta}}_{n,m}\rightarrow\hat{\tilde{\bar{\beta}}}_{n,m}$. These
replacements allow us, via the basic formula

${d\sigma}=\sum_{i,j}\int
dx_{1}dx_{2}{F_{i}(x_{1})F_{j}(x_{2})}d\hat{\sigma}_{\rm res}(x_{1}x_{2}s),$
(4)

to proceed with connection to MC@NLO as explained in Ref. [7].

Our (BS, BFLW) recent applications [21, 22] make comparisons between the LHC
data on $W+n$ jets, n=1,2,3, and the exact NLO ME matched QCD parton shower
predictions in the MG5_aMC@NLO framework with the parton shower realized via
Herwig6.5 and Herwiri1.031 respectively for the unimproved and IR-improved
results. We illustrate such results in Figs. 2 and 3 for the

(a)(b) Figure 2: Comparison of CMS 8TeV cms energy $W+\geq 1$ jet (a) and
$W+\geq 2$ jets (b) data for the respective leading jet and second leading jet
$p_{T}$ distributions and the IR-improved(herwiri) and unimproved (herwig)
exact NLO ME matched parton shower predictions. (a)(b) Figure 3: Comparison of
CMS 8TeV cms energy $W+\geq 3$ jets (a) and $W+\geq 1$ jet (b) data for the
respective 3rd-leading jet $p_{T}$ and $H_{T}$ distributions and the IR-
improved(herwiri) and unimproved (herwig) exact NLO ME matched parton shower
predictions.

CMS [23] 8 TeV cms energy $W+\geq n$ jet data, n=1, 2, 3. Consistent with
Refs. [7], the IR-improved results are closer to the data for low
$p_{T}(H_{T})$.

Two of us (BFLW and SAY) have investigated the effect of IR-improvement on the
discovery reach of a standard candle process such as single $Z/\gamma^{*}$
production at the FCC using the predicted inclusive cross section for
$Z/\gamma^{*}$ as a function of $p_{T,min}$ 333This observable was suggested
by M.L. Mangano, private communication, 2016. which is explained and shown in
Fig. 6 in Ref. [24]. In the latter figiure we plot predictions for the
following: MG5_aMC@NLO/A, A= Herwig6.5, Herwiri1.031, Herwig++ and Pythia8,
all with the common renormalization and factorization scale of $M_{Z}$/2 and
all with the common renormalization and factorization scale of $H_{T}$/2
(denoted by ’UNFIX’ in the legend in the figure<EMAIL_ADDRESS>and
fixed order NLO both with the common renormalization and factorization
$M_{Z}$; and, fixed order NLO with the common renormalization and
factorization scale $H_{T}$/2. Here, $H_{T}$ is the sum of the transverse
masses of the final state particles – see Ref. [24] for the remaining details.
Our results show that the dynamical scale choice makes a big difference in the
expectations, the fixed-order NLO results agree, as it is expected, with the
<EMAIL_ADDRESS>results for both of the scale choices, and the IR-
improved and unimproved predictions agree within the statistical
uncertainties.

Finally, in Refs. [25, 26, 27] two of us (BFLW and SAY), in collaboration with
S. Jadach and Z. Was, have analyzed the effects of the exact ${\cal
O}(\alpha^{2}L)$ CEEX EW corrections in ${\cal KK}$MC-hh on the analysis of
the $Z$ observables used in the ATLAS $M_{W}$ measurement in Ref. [28]. We
find new effects that should be considered at or above the per mille level.
For example, the new modulation we see in the lepton $p_{T}$ spectrum seems to
match the trends in the data vs theory comparisons shown in Figs. 15 a and b
in Ref. [28]. More specifically, we have found new effects sensitive to the
transverse degrees of freedom in the ISR radiation as well as significant ISR
effects in observables primarily sensitive to collinear ISR degrees of
freedom, as we illustrate here in Table 1.

Table 1: Illustrations of ISR and IFI Effects

Numerical Results

| No ISR | LuxQED ISR |  ${\cal KK}$MC-hh ISR |  ISR$-$no ISR | With IFI |  %(IFI $-$ no IFI)
---|---|---|---|---|---|---
Uncut $\sigma$ | 939.86(1) pb | 944.04(1) pb | 944.99(2) pb | 0.546(2)% | 944.91(2) pb |  $-0.0089(4)$%
Cut $\sigma$ | 439.10(1) pb | 440.93(1) pb | 442.36(1) pb | 0.742(3)% | 422.33(1) pb |  $-0.0070(5)$%
$A_{\rm FB}$ | 0.01125(2) | 0.01145(2) | 0.1129(2) | $(3.9\pm 2.8)\times 10^{-5}$ | 0.01132(2) | $(2.9\pm 1.1)\times 10^{-5}$
$A_{4}$ | 0.06102(3) | 0.06131(3) | 0.06057(3) | $(-4.4\pm 0.5)\times 10^{-4}$ | 0.06102(3) | $(4.5\pm 0.3)\times 10^{-4}$

The ISR and IFI effects on $A_{FB}$ is of order $10^{-5}$ while the effect on
$A_{4}$ is of order $10^{-4}$

in ${\cal KK}$MC-hh. LuxQED gives an ISR effect on the order of $10^{-4}$ for
both $A_{FB}$ and $A_{4}$.

We are extending the analysis in Refs. [25, 26, 26] to IR-improved showers
with the new MC ${\cal KK}$MC-hh/Herwiri1.031 as explained and illustrated in
Ref. [24].

## Acknowledgments

This work was partially supported by The Citadel Foundation. We thank the INP,
Krakow, PL for computational resources. One of us (BFLW) thanks Prof. Gian
Giudice for the support and kind hospitality of the CERN TH Department.

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