French to English Mathieu 1873 Translation

_On the five-fold transitive function of 24 quantities;
Pages 25-46
_

Original Document Written By Émile Mathieu

Project Begun on 6th November 2021

Translation Written by
Yiming Bing, Bright Hu, Ronni Hu, Rhianna Kho, Max Lau, Rhianna Li, Stefan Lu,
Finn Mcdonald, Dr Michael Sun, Gavin Trann, Nicholas Wolfe, Joshua Yao, Leon
Zhou, Nathan Zhou*

The link to the original document in French is
http://sites.mathdoc.fr/JMPA/PDF/JMPA_1873_2_18_A2_0.pdf

In my Memoir _on the functions of several quantities_ published in volume VI
of this Journal, in 1861, I declared (p. 274) that I had a five-fold
transitive function of 24 quantities and gave, at the same time, the number of
its distinct values.

I had noted two substitutions which leave it invariant, characterizing it
completely, so it was easy to verify its existence; but only having this and
knowing the number of distinct values sheds no light on the formation of this
function. Since then, no one has attempted a proof and I now propose to prove
here it actually exists, effectively, and to show how I came to discover it.

I had not published this function in the cited Memoir because the method I was
to use lacked the sharpness and elegance of my other results. However, this
method will still be of interest to those who can imagine the difficulty of
discovering such a function.

## Indication of research process

1. Let $p$ be a prime number and let $x_{0},x_{1},x_{2},\dots,x_{p-1}$ be $p$ quantities, such that, $x_{m}=x_{n}$ whenever $m$ is congruent to $n$ modulo $p$. There are transitive functions of these quantities which are only invariant by substitutions of the form

$(x_{z},x_{az+b});$

we omit these functions, the symmetric function and the two valued function
from further discussions about transitive functions of $p$ letters111It seems
that a transitive function on $p$-letters could somehow correspond to the
modern phrase a group of permutations acting transitively on a $p$-element set
because they are well-known.

With this understood, consider a transitive function of a prime number $p$ of
quantities, and suppose that $\frac{p-1}{2}$ is also a prime number $q$; I
demonstrated in the Memoire cited (Chap. IV), that, by designating the $p$
quantities in a suitable order by $x_{0},x_{1},\dots,x_{p-1}$ the function is
invariant not only by the cyclic substitution

(A)

$(x_{0},x_{1},x_{2},\dots\,x_{p-1}),$

but also by the substitution

(B)

$(x_{1},x_{g^{2}},x_{g^{4}},\dots,x_{g^{p-3}})(x_{g},x_{g^{3}},x_{g^{5}},\dots,x_{g^{p-2}}),$

where $g$ is a generator modulo $p$; consequently, it is invariant under the
$\frac{p(p-1)}{2}$ derived substitutions, included in the formula

$(z,a^{2}z+b),$

$a$ and $b$ being arbitrary.

Now here is the principle on which I relied to determine the function in
question.

_Let us represent the substitution (B) by_
(C)
$(x_{0}^{\prime},x_{1}^{\prime},x_{2}^{\prime},\dots,x_{q-1}^{\prime})(x_{0}^{\prime\prime},x_{1}^{\prime\prime},x_{2}^{\prime\prime},\dots,x_{q-1}^{\prime\prime})$

_where_

$x^{\prime}_{0}=x_{1},x^{\prime}_{1}=x_{g^{2}},x^{\prime}_{2}=x_{g^{4}},\dots,x^{\prime}_{q-1}=x_{g^{p}-2};$

_then, if the second cycle of (B) is similarly identified with the second
cycle of (C), the transitive function is necessarily invariant under a regular
substitution of $p-3$ quantities of the form_
(D)
$(x^{\prime}_{z},x^{\prime}_{y^{u}z})(x^{\prime\prime}_{z},x^{\prime\prime}_{y^{u}z})$

_where $y$ is a generator modulo $q$, $u$ a divisor of $q-1$, smaller than
$q-1$, and the indices being taken according to the modulus $q$._

Before applying this principle to the search for a $23$-letter transitive
function, let’s apply it to the determination of the $7$\- and $11$-letter
transitive functions, since $\frac{7-1}{2}$ and $\frac{11-1}{2}$ are prime
numbers.

## Transitive functions of 7 and 11 letters

2. From what we have just recalled, any transitive function of the $7$ quantities $x_{0},x_{1},x_{2},\dots,x_{6}$ can be considered as invariant under the substitutions

$(z,z+1)$ and $(z,3^{2}z)$

or by substitutions

$(e)\quad(x_{0},x_{1},x_{2},x_{3},x_{4},x_{5},x_{6}),$
$(f)\quad(x_{1},x_{2},x_{4})(x_{3},x_{6},x_{5}).$

Let’s put these last substitutions in the form

$(g)\quad(x^{\prime}_{0},x^{\prime}_{1},x^{\prime}_{2})(x^{\prime\prime}_{0},x^{\prime\prime}_{1},x^{\prime\prime}_{2})$

and we will have the substitution (D)

$(h)\quad(x^{\prime}_{1},x^{\prime}_{2})(x^{\prime\prime}_{1},x^{\prime\prime}_{2})$

To identify (f) and (g), we will start by doing

$x^{\prime}_{0}=x_{1}$, $x^{\prime}_{1}=x_{2}$, $x^{\prime}_{2}=x_{4}$,

and we will have, by considering the second cycles, the following three
hypotheses:

$1^{\circ}$
$x^{\prime\prime}_{0}=x_{3},x^{\prime\prime}_{1}=x_{6},x^{\prime\prime}_{2}=x_{5};$

$2^{\circ}$
$x^{\prime\prime}_{0}=x_{6},x^{\prime\prime}_{1}=x_{5},x^{\prime\prime}_{2}=x_{3};$

$3^{\circ}$
$x^{\prime\prime}_{0}=x_{5},x^{\prime\prime}_{1}=x_{3},x^{\prime\prime}_{2}=x_{6}.$

And, according to that, the substitution (h) will be one of the following
three:

$1^{\circ}$ $(x_{2}x_{4})(x_{6}x_{5}),$

$2^{\circ}$ $(x_{2}x_{4})(x_{5}x_{3}),$

$3^{\circ}$ $(x_{2}x_{4})(x_{3}x_{6}).$

Any of the substitutions $1^{\circ},2^{\circ},3^{\circ}$, combined with (e)
and (f), gives a system of $4\times 6\times 7$ conjugated substitutions, which
belongs to a twice transitive function which has $\frac{2\cdot 3\cdot 4\cdot
5}{4}=30$ values. These three systems of conjugated substitutions or the three
corresponding functions differ only in the way of designating the $7$
quantities.

Consider, for example, the first of these three functions. From the
substitutions (f) and $1^{\circ}$, we immediately see that it is considered as
a function of the $6$ quantities $x_{1},x_{2},x_{3},x_{4},x_{5},x_{6}$ it is
not only invariable by $1^{\circ}$, but also by

$(x_{2}x_{6})(x_{4}x_{5});$

the function is therefore transitive with respect to
$x_{2},x_{4},x_{6},x_{3}$, and it has indeed $30$ values.

According to a general theorem, a transitive function being necessarily
invariable by the substitution (f), it suffices, to characterise the current
function, to say that it is invariant by the substitutions ($e$) and
$1^{\circ}$.

Let $x_{\infty}$ denote an eighth quantity; there is a function of the $8$
letters $x_{0},x_{1},\dots,x_{6},x_{\infty}$ which, considered as a function
of the first $7$ letters, is identical to the previous function, so that it is
invariant by the substitutions ($e$) and $1^{\circ}$; it is, moreover,
invariant by all the substitutions

$\left(z,\frac{Az+B}{Cz+D}\right),$

for which $AD$-$BC$ is a quadratic residue of $7$; it will be characterised by
the substitutions ($e$) and $1^{\circ}$, and any of the latter substitutions
containing $x_{\infty}$, for example the following:

$(x_{z}x_{-\frac{1}{z}})=(x_{0}x_{\infty})(x_{1}x_{6})(x_{2}x_{3})(x_{4}x_{5}).$

This three times transitive function of $8$ quantities is enclosed in a family
of twice transitive functions of $p^{v}$ quantities, which become three times
transitive for $p=2$ and which have $\frac{1\cdot 2\dots
p^{v}}{p^{v}(p^{v}-1)\dots(p^{v}-p^{v-1})}$ values. I have given this family
of functions in Chapter III of the Memoir cited, and I have given it a very
elegant method of forming them. I have also shown that these functions should
be used as solvers of equations whose degree is a power of a prime number.
(See _Memoir on the resolution of equations_ , in the _Annals of Pure and
Applied Mathematics_ , t. IV, 1862.)

3. Let us apply the same method to the determination of the transitive functions of $11$ letters.

First, any transitive function of $11$ quantities can be considered as
invariant under the substitutions

$(z,z+1),\qquad(z,2^{2}z),$

or by substitutions

$(k)\quad(x_{0},x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8},x_{9},x_{10}),$
$(l)\quad(x_{1},x_{4},x_{5},x_{9},x_{3})(x_{2},x_{8},x_{10},x_{7},x_{6}).$

Let us represent the last substitution by

$(m)\quad(x^{\prime}_{0},x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3},x^{\prime}_{4})(x^{\prime\prime}_{0},x^{\prime\prime}_{1},x^{\prime\prime}_{2},x^{\prime\prime}_{3},x^{\prime\prime}_{4})$

identify ($m$) with ($l$) starting by putting

$x^{\prime}_{0}=x_{1},x^{\prime}_{1}=x_{4},x^{\prime}_{2}=x_{5},x^{\prime}_{3}=x_{9},x^{\prime}_{4}=x_{3},$

and we will have to make one of the following five assumptions:

$1^{\circ}$
$x^{\prime\prime}_{0}=x_{2},\>x^{\prime\prime}_{1}=x_{8},x^{\prime\prime}_{2}=x_{10},x^{\prime\prime}_{3}=x_{7},x^{\prime\prime}_{4}=x_{6};$

$2^{\circ}$
$x^{\prime\prime}_{0}=x_{8},\>x^{\prime\prime}_{1}=x_{10},x^{\prime\prime}_{2}=x_{7},x^{\prime\prime}_{3}=x_{6},\>x^{\prime\prime}_{4}=x_{2};$

$3^{\circ}$
$x^{\prime\prime}_{0}=x_{10},x^{\prime\prime}_{1}=x_{7},x^{\prime\prime}_{2}=x_{6},x^{\prime\prime}_{3}=x_{2},x^{\prime\prime}_{4}=x_{8};$

$4^{\circ}$
$x^{\prime\prime}_{0}=x_{7},\>x^{\prime\prime}_{1}=x_{6},x^{\prime\prime}_{2}=x_{2},x^{\prime\prime}_{3}=x_{8},x^{\prime\prime}_{4}=x_{10};$

$5^{\circ}$
$x^{\prime\prime}_{0}=x_{6},\>x^{\prime\prime}_{1}=x_{2},x^{\prime\prime}_{2}=x_{8},x^{\prime\prime}_{3}=x_{10},x^{\prime\prime}_{4}=x_{7}.$

The substitution

$(D)\quad(x^{\prime}_{z}x^{\prime}_{y^{u}z})(x^{\prime\prime}_{z}x^{\prime\prime}_{y^{u}z}),$

whose indices are taken according to modulus $5$, will be, taking $y=2$,
depending on whether $u=2$ or $u=1$,

$(n)\quad(x^{\prime}_{1},x^{\prime}_{4})(x^{\prime}_{2}x^{\prime}_{3})(x^{\prime\prime}_{1}x^{\prime\prime}_{4})(x^{\prime\prime}_{2}x^{\prime\prime}_{3}),$

or

$(p)\quad(x^{\prime}_{1}x^{\prime}_{2}x^{\prime}_{4}x^{\prime}_{3})(x^{\prime\prime}_{1}x^{\prime\prime}_{2}x^{\prime\prime}_{4}x^{\prime\prime}_{3})$

.

Substitution ($n$) is the second power of substitution ($p$). So the system of
conjugate substitutions deduced from ($k$) and ($p$) obviously contains the
system of conjugate substitutions deduced from ($k$) and ($n$); therefore, if
the system deduced from ($k$) and ($p$) gives a transitive function of $11$
letters, we are sure that the other system also gives one, while the converse
is not obvious.

So let us start by dealing with the conjugate substitution system deduced from
$(k)$ and $(n)$. Based on the five possible assumptions, $(n)$ is one of the
following five substitutions:

$1^{\circ}$ $(x_{4}x_{3})(x_{5}x_{9})(x_{8}x_{6})(x_{10}x_{7}),$

$2^{\circ}$ $(x_{4}x_{3})(x_{5}x_{9})(x_{10}x_{2})(x_{7}x_{6}),$

$3^{\circ}$ $(x_{4}x_{3})(x_{5}x_{9})(x_{7}x_{8})(x_{6}x_{2}),$

$4^{\circ}$ $(x_{4}x_{3})(x_{5}x_{9})(x_{6}x_{10})(x_{2}x_{8}),$

$5^{\circ}$ $(x_{4}x_{3})(x_{5}x_{9})(x_{2}x_{7})(x_{8}x_{10}).$

The substitutions $1^{\circ},3^{\circ},5^{\circ}$ are not suitable, that is to
say that the system of conjugate substitutions derived from $(k)$ and from one
of these three substitutions is nothing other than the $\frac{1\cdot 2\cdot
3\dots 11}{2}$ substitutions which leave the function of the $11$ letters
which has two values only. We have to show how we can recognize it.

This is very easy if we are willing to make use of this principle, which has
served me many times in my research, but not in my demonstrations: it is that
a transitive function of a prime number $p$ quantities cannot be invariant by
a substitution taking place on less than $\frac{p+1}{2}$ quantities.222In a
Memoir presented to the Academy of Sciences on May 31, 1858, I demonstrated
the following theorem, which could also serve the goal proposed here: _If a
transitive function of n letters is transitive with respect to a certain
number a of its letters which is not greater than $\frac{n}{2}$, these n
letters can be divided into groups each composed of the same number of
letters, so that the function is transitive with respect to these groups,
which are transitive._ But, although I have never doubted the correctness of
this theorem, the proof not having fully satisfied me, I have never forgotten
it.

Consider, for example, the system deduced from $(k)$ and $1^{\circ}$. Let us
do $1^{\circ}$ once, then $(k)$ twice, and form the table.

$0,1,2,3,4,5,6,7,8,9,10$

$0,1,2,4,3,9,8,10,6,5,7$

$2,3,4,6,5,0,10,1,8,7,9.$

We will have made, ultimately, the substitution $(0,2,4,5)(1,3,5,10,9,7)$,
which is part of the system of conjugate substitutions; do this last
substitution six times, we will have the substitution

$(q)\quad(0,4)(2,5),$

which is also part of this system, and which has only four letters; therefore,
according to the above principle, substitution $1^{\circ}$ should be rejected.

If we do not want to use this principle, here is how we can note that there
are no transitive functions invariable by (k) and $1^{\circ}$. We will deduce
from these two substitutions some others which do not not contain the same
letter, $x_{0}$ for example. Most often, we will be able to be satisfied with
two substitutions which do not contain $x_{0}$; we will notice that the
function is transitive with respect to the letters remaining
$x_{1},x_{2},\dots,x_{10}$. From the substitutions obtained on these $10$
quantities, let us deduce some others not containing $x_{1}$, and note that
the function, according to these last substitutions, is transitive with
respect to the $9$ letters $x_{2},x_{3},\dots,x_{10}$. We then determine
derived substitutions not containing $x_{2}$, and so on. Continuing thus, we
would prove that the function is transitive with respect to $10$ letters, then
to $9$ takes from the previous ones, then to $8$ takes from the latter, then
to $7$, finally to $6$ letters.

However, a function of $n$ letters cannot be more than $\frac{n}{2}$ times
transitive when it has more than two values (see our Memoir _On the Number of
Values that a Function can Acquire_ , t. V of this Journal, 1860, p. 18);
therefore the considered function cannot have more than two values.

This method is very long, but it can be simplified a lot. Indeed, in the
current case for example, the function is invariant by the substitution (q).
So, as soon as we recognize that the function is four times transitive, we can
conclude that it is transitive with respect to any $4$ letters, and that it
has only two values, since instead of the $4$ letters of the substitution
$(q)$ we can bring any $4$ letters into the function.
It will thus be recognized that substitution $1^{\circ}$ must be rejected, and
it is the same for substitutions $3^{\circ}$ and $5^{\circ}$. It remains to
consider the two substitutions $2^{\circ}$ and $4^{\circ}$ which are suitable.
The function characterized by $(k)$ and $2^{\circ}$ is twice transitive, and,
considered as a function of the 9 quantities $x_{2},x_{3},\dots,x_{10}$, it is
invariant only by substitutions derived from

$(r)\quad(2,4,7)(3,10,6)(5,8,9)$

and substitution $2^{\circ}$ or

$(4,3)(5,9)(10,2)(7,6);$

the substitutions derived from these last two are the second power of $(r)$
and these other two

$(7,10)(8,5)(6,4)(2,3),$

$(2,6)(9,8)(3,7)(4,10).$

So, by swapping the $9$ quantities $x_{2},x_{3},\dots,x_{10}$, we get $6$
equal values of the function, and this function, which is twice transitive,
has $\frac{1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8\cdot
9}{6}=60,480$ values. It was first given by M. Kronecker.

The transitive function characterised by (k) and $4^{\circ}$ is twice
transitive, and, considered as a function of the $9$ letters
$x_{2},x_{3},\dots,x_{10}$, it is invariable by the substitution

$(7,4,3)(2,6,9)(8,5,10),$

the substitution $4^{\circ}$ and those derived from it. It is easy to see that
this function differs from the preceding one only by the way of designating
the $11$ quantities which it contains.

Thus, from the examination of the substitution (n), it follows that _there
exists a twice transitive function of eleven letters which has $60480$ values
and characterized by the substitutions_

$(E)\quad(x_{0},x_{1},x_{2},\dots,x_{10}),$
$(H)(x_{4},x_{3})(x_{5},x_{9})(x_{10},x_{2})(x_{7},x_{6}).$

_By adding any of the substitutions_

$(z,\frac{Az+B}{Cz+D})$

_for which AD-BC is a quadratic residue of 11 and containing $x_{\infty}$, for
example by_

$(z,-\frac{1}{z})=(\,0,\infty\,)\;(\,1,10)\;(\,2,5)\;(\,3,7)\,(\,4,8)\,(\,6,9\,)$

_which we have the three substitutions which characterize three times
transitive function of the twelve quantities,
$x_{\infty},x_{0},x_{1},\dots,x_{10}$_, which, considered as a function of the
last eleven quantities, is identical to the twice transitive function which
precedes.

4. Now let’s consider the substitution (_p_). The first, third and fifth identifications, having to be rejected in the examination of the substitution (_n_), all the more so now; it is therefore necessary to try only the second and the fourth identification, which gives, for the substitution (_p_), one of these two formulas

$(s)\quad(x_{4},x_{5},x_{3},x_{9})(x_{10},x_{7},x_{2},x_{6}),$
$(t)\quad(x_{4},x_{5},x_{3},x_{9})(x_{6},x_{10},x_{2},x_{8}),$

There exists a function transitive of 11 letters, characterised by (_k_) and
(_s_) or by (_k_) and (_t_). Since the substitutions (_k_) and (_s_) give the
same system of conjugated substitutions as (_k_) and (_t_), let us consider
only the first case.

The function characterised by (_k_) and (_s_) is evidently invariable by the
whole system of conjugate substitutions of the function of Mr Kronecker; as a
consequence, it is invairable by the substitution (_r_), and, adding it to
(_l_) and (_s_) we have all three substitutions.
(L) $(x_{1},x_{4},x_{5},x_{9},x_{3})(x_{2},x_{8},x_{7},x_{6})$
(R) $(x_{2},x_{4},x_{7})(x_{3},x_{1}0,x_{6})(x_{5},x_{8},x_{9})$
(S) $(x_{4},x_{5},x_{3},x_{9})(x_{1}0,x_{7},x_{2},x_{6})$

We can recognize that the system of conjugate substitutions derived from these
three substitutions, which are carried out on the $10$ quantities
$x_{2},x_{2},\dots,x_{10}$ is three times transitive of $11$ letters which has
$7!$ values

Thus, from the examination of the substitution (p), it follows that _there is
a four times transitive function $\gamma$ of eleven letters which has
$7!=5040$ values characterised by the substitutions_

$(x_{0},x_{1},x_{2},\dots,x_{10},)$
$(x_{4},x_{5},x_{3},x_{9})(x_{10},x_{7},x_{2},x_{6}),$

_in whose system of conjugate substitutions contains the system of
substitutions of the twice transitive function characterised by the
substitutions (E) and (H). if to these two substitutions we add the
substitution $(z,-\frac{1}{z}),$ we have the three substitutions which
characterise a five times transitive function of twelve quantities_.

I present the discovery of this five-fold transitive function in a very
different way at the end of Chapter II of my Memoir _on the functions of
several quantities_. It is useful to compare the form in which it is presented
here to the one in which we first made it known.

Let $\omega$ be a root of an irreducible quadratic taken with respect to
modulus 3, for example a root of the congruence

$\omega^{2}+2\omega+2\equiv 0(\mathrm{mod}\quad 3);$

all the roots of

$x^{3^{2}}\equiv x(\mathrm{mod}\quad 3)$

are equal to

$0,1,2,\omega,1+\omega,2+\omega,2\omega,1+2\omega,2+2\omega;$

the quantity chosen for $\omega$ is, moreover, a primitive root of the
congruence

$x^{3^{2}-1}\equiv 1(\mathrm{mod}\quad 3);$

and we will distinguish the 8 roots of this congruence into two categories:
the first, which contains the odd powers of $\omega$,

$\omega,\omega^{2}\equiv 1+2\omega,\quad\omega^{5}\equiv
1+2\omega,\quad\omega^{7}\equiv 1+\omega$

and which we call _quadratic non-residues_ ; the second, which contains the
even powers of $\omega$,

$\omega^{2}\equiv 1+\omega,\omega^{4}\equiv 2,\omega^{6}\equiv
2+2\omega,\omega^{8}\equiv 1$

which we call _quadratic residues_.

According to this, let us change notations and denote the ten quantities

$x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{9},x_{10},x_{8},x_{1}$

repectively by

$\gamma_{\omega},\gamma_{1+\omega},\gamma_{2+2\omega},\gamma_{1+2\gamma},\gamma_{2},\gamma_{1},\gamma_{2+\omega},\gamma_{2\omega},\gamma_{0},\gamma_{\infty}$

then the three substitutions (L), (R), (S) become:

$(\gamma_{\infty},\gamma_{2+2\omega},\gamma_{1+2\omega},\gamma_{2+\omega},\gamma_{1+\omega})(\gamma_{\omega},\gamma_{0},\gamma_{2\omega},\gamma_{1},\gamma_{2}),$
$\gamma_{\omega},\gamma_{2+2\omega},\gamma_{1})(\gamma_{1+\omega},\gamma_{2\omega},\gamma_{2})(\gamma_{1+2\omega},\gamma_{0},\gamma_{2+\omega}),$
$(\gamma_{2+2\omega},\gamma_{1+2\omega},\gamma_{1+\omega},\gamma_{2+\omega})(\gamma_{2\omega},\gamma_{1},\gamma_{\omega},\gamma_{2}),$

and can be put in this form

$(z,\frac{z+2\omega}{(1+\omega)z+1}),$ $(z,z+2+\omega),$ $(z,\omega{z^{3}});$

We can easily conclude that all the substitutions of 8, 9 and 10 of the
quantities represented by the notion $\gamma$ are included in the two formulas

$(\gamma_{z},\gamma_{\frac{Az+B}{Cz+D}}),(\gamma_{z},\gamma_{\frac{A_{1}z^{3}+B_{1}}{C_{1}z^{3}+D_{1}}}),$

where $AD-BC$ is quadratic residue and $A_{1}D_{1}-B_{1}C_{1}$ non-residue,
and these two formulas represent a system of $8\dot{9}\dot{1}0$ conjugated
substitutions, three times transitive.

We have relied on this theorem that any transitive function of eleven
quantities is invariant by the substitution (l) or

$(z,2^{2}z);$

but this theorem does not prove that such a function cannot be invariant by
the substitution $(z,2z)$ or
(u)

$(x_{1}x_{2}x_{4}x_{8}x_{5}x_{10}x_{9}x_{7}x_{3}x_{6}),$

of which the preceding is a second power. To recognize if such a function can
exist, we will combine this substitution with the two substitutions (E) and
(H), which characterise the twice transitive function of 11 letters, and we
will see that the system of conjugate substitutes which s’ would deduce from
it would be none other than the $1.2.3\dots 11$ substitutions which leave the
symmetric function invariant. It follows that there are no transitive
functions of $11$ letters invariant by the substitution (u), or more generally
by no even substitution. (We call _pair substitution_ the one that changes the
function that has two values.)

## Functions acting transitively on $23$ elements

5. Since the number $2$3 is a prime number, we can apply the same method to the determination of the transitive functions of $23$ quantities.

From what we know, any transitive function of $23$ quantities
$x_{0},x_{1},x_{2},\cdots,x_{22}$ can be considered invariant by the
substitutions

$(z,z+1)\;\text{and}\;(z,5^{2}z),$

since $5$ is the primitive root of $23$, or by substitutions

(A)$(x_{0},x_{1},x_{2},x_{3},\dots,x_{22})$

$\text{(B)}\begin{cases}(x_{1},x_{2},x_{4},x_{8},x_{16},x_{9},x_{18},x_{13},x_{3},x_{6},x_{12}),\\\
(x_{5},x_{10},x_{20},x_{17},x_{11},x_{22},x_{21},x_{19},x_{15},x_{7},x_{14}).\\\
\end{cases}$

Let’s put this last substitution in this form

$\text{(C)}\begin{cases}(x^{\prime}_{0},x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3},x^{\prime}_{4},x^{\prime}_{5},x_{6}^{\prime},x^{\prime}_{7},x^{\prime}_{8},x^{\prime}_{9},x^{\prime}_{10})\\\
(x^{\prime\prime}_{0},x^{\prime\prime}_{1},x^{\prime\prime}_{2},x^{\prime\prime}_{3},x^{\prime\prime}_{4},x^{\prime\prime}_{5},x^{\prime\prime}_{6},x^{\prime\prime}_{7},x^{\prime\prime}_{8},x^{\prime\prime}_{9},x^{\prime\prime}_{10})\end{cases}$

and let $x^{\prime}_{0},x^{\prime}_{1},\dots,x^{\prime}_{10}$ be equal to the
quantities of the same row of the first cycle of (B), so that we have

$x^{\prime}_{0}=x_{1},x^{\prime}_{1}=x_{2},x^{\prime}_{2}=x_{4},x^{\prime}_{3}=x_{8},x^{\prime}_{4}=x_{16},x^{\prime}_{5}=x_{9},x^{\prime}_{6}=x_{18},x^{\prime}_{7}=x_{13},x^{\prime}_{8}=x_{3},x^{\prime}_{9}=x_{6},x^{\prime}_{10}=x_{12};$

but the identification of the second cycle of (C) with the second cycle of (B)
can be done in eleven different ways:

$1^{\circ}\begin{cases}x^{\prime\prime}_{0}=x_{5},x^{\prime\prime}_{1}=x_{10},x^{\prime\prime}_{2}=x_{20},x^{\prime\prime}_{3}=x_{17},x^{\prime\prime}_{4}=x_{11},x^{\prime\prime}_{5}=x_{22},\\\
x^{\prime\prime}_{6}=x_{21},x^{\prime\prime}_{7}=x_{19},x^{\prime\prime}_{8}=x_{15},x^{\prime\prime}_{9}=x_{7},x^{\prime\prime}_{10}=x_{14};\end{cases}$
$2^{\circ}\begin{cases}x^{\prime\prime}_{0}=x_{10},x^{\prime\prime}_{1}=x_{20},x^{\prime\prime}_{2}=x_{17},x^{\prime\prime}_{3}=x_{11},x^{\prime\prime}_{4}=x_{22},x^{\prime\prime}_{5}=x_{21},\\\
x^{\prime\prime}_{6}=x_{19},x^{\prime\prime}_{7}=x_{15},x^{\prime\prime}_{8}=x_{7},x^{\prime\prime}_{9}=x_{14},x^{\prime\prime}_{10}=x_{5}\par\end{cases}$

……………………………………………………………………………………………………………….

For the substitution (D) of n°1, we have to take the substitution

$(T)\begin{cases}(x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{4},x^{\prime}_{8},x^{\prime}_{5},x^{\prime}_{10},x^{\prime}_{9},x^{\prime}_{7},x^{\prime}_{3},x^{\prime}_{6},)\\\
(x^{\prime\prime}_{1},x^{\prime\prime}_{2},x^{\prime\prime}_{4},x^{\prime\prime}_{8},x^{\prime\prime}_{5},x^{\prime\prime}_{10},x^{\prime\prime}_{9},x^{\prime\prime}_{7},x^{\prime\prime}_{3},x^{\prime\prime}_{6},)\end{cases}$

composed of two cycles of 10 quantities, or its fifth power, or its second
power. Its fifth power is

$(x^{\prime}_{1}x^{\prime}_{10})(x^{\prime}_{2}x^{\prime}_{9})(x^{\prime}_{3}x^{\prime}_{8})(x^{\prime}_{4}x^{\prime}_{7})(x^{\prime}_{5}x^{\prime}_{6}),$
$(x^{\prime\prime}_{1}x^{\prime\prime}_{10})(x^{\prime\prime}_{2}x^{\prime\prime}_{9})(x^{\prime\prime}_{3}x^{\prime\prime}_{8})(x^{\prime\prime}_{4}x^{\prime\prime}_{7})(x^{\prime\prime}_{5}x^{\prime\prime}_{6}),$

and, after having applied one of the $11$ identifications, we can verify that
it must be rejected. A fortiori, the substitution (T) must be set aside.

It therefore remains to consider the second power of (T), which is

$(x_{1}^{\prime}x_{4}^{\prime}x_{5}^{\prime}x_{9}^{\prime}x_{3}^{\prime})(x_{2}^{\prime}x_{8}^{\prime}x_{10}^{\prime}x_{7}^{\prime}x_{6}^{\prime}),$
$(x_{1}^{\prime\prime}x_{4}^{\prime\prime}x_{5}^{\prime\prime}x_{9}^{\prime\prime}x_{3}^{\prime\prime})(x_{2}^{\prime\prime}x_{8}^{\prime\prime}x_{10}^{\prime\prime}x_{7}^{\prime\prime}x_{6}^{\prime\prime}),$

If we apply our original identifications we get the substitution

$\text{(U)}\begin{cases}(x_{2},x_{16},x_{9},x_{5},x_{8})(x_{4},x_{3},x_{12},x_{13},x_{18})\\\
(x_{10},x_{11},x_{22},x_{7},x_{17})(x_{20},x_{15},x_{14},x_{19},x_{21})\end{cases}$

which provides a function that we will examine. As for the 10 other
identifications, they won’t give new functions. We take functions generated by
$(A)and(U)$ which omits $x_{0}$ and hence act transitively on $22$ quantities,
namely $x_{1},x_{2},x_{3},\dots,x_{22}$; we will then form substitutions
which, besides $x_{0}$, also omits $x_{1}$ and get a transitive action on 21
quantities $x_{2},x_{3},x_{4},\dots,x_{22}$. We can also omit
$x_{0},x_{1},x_{5}$ (from (U)) to get functions acting transitively on $20$
quantities. These functions therefore act four fold transitively.

Finally, consider the substitutions which do not contain $x_{0}$, $x_{1}$,
$x_{5}$, and a fourth quantity, for example $x_{2}$. We derive: $1^{\circ}$
substitution

$\text{(Z)}\begin{cases}(4,19)(15,12)(16,7)(10,8)(18,9)(6,13)(11,20)(21,22),\\\
(4,16)(19,7)(15,10)(12,8)(18,11)(9,20)(6,21)(13,22),\\\
(4,18)(19,9)(15,6)(12,13)(16,11)(7,20)(10,21)(8,22),\\\
(4,12)(19,15)(16,8)(7,10)(18,13)(9,6)(11,22)(20,21),\end{cases}$

according to which, as we can see at first glance, the function is transitive
with respect to the 16 quantities which it has as indices:
$4,6,7,8,9,10,11,12,13,15,16,18,19,20,21.22;$

$2^{\circ}$ substitution
$(3,14,17)(7,19,21)(13,15,12)(4,10,11)(9,8,16)(20,18,22).$

We can conclude that there are $16\times 3$ substitutions which do not contain
$x_{0},x_{1},x_{5},x_{2}$ and leave the function invariant, given the four
times transitive function of $23$ quantities has $\frac{1\cdot 2\cdot 3\cdot
4\cdots 19}{16\times 3}$ values. We are now going to make a very curious
remark: it is that, if we represent the substitution ($U$) cycle by cycle,by

$\left(\begin{array}[]{l}\left(\gamma_{0}\gamma_{1},\gamma_{2}\gamma_{3}\gamma_{4}\right)\left(\gamma_{0}^{\prime}\gamma_{1}^{\prime}\gamma_{2}^{\prime}\gamma_{3}^{\prime}\gamma_{4}^{\prime}\right)\\\
\left(\gamma_{0}^{\prime\prime}\gamma_{1}^{\prime\prime}\gamma_{2}^{\prime\prime}\gamma_{3}^{\prime\prime}\gamma_{4}^{\prime\prime}\right)\left(\gamma_{0}^{\prime\prime\prime}\gamma_{1}^{\prime\prime\prime}\gamma_{2}^{\prime\prime\prime}\gamma_{3}^{\prime\prime\prime}\gamma_{4}^{\prime\prime\prime}\right)\end{array}\right)$

the function is invariant by the substitution
$\left(\gamma_{z}\gamma_{4z}\right)$ with the indices being taken according to
$\mod{5}$, i.e. by

$\left\\{\begin{array}[]{l}\left(\gamma_{1}\gamma_{4}\right)\left(\gamma_{2}\gamma_{3}\right)\left(\gamma_{1}^{\prime}\gamma_{4}^{\prime}\right)\left(\gamma_{2}^{\prime}\gamma_{3}^{\prime}\right)\\\
\left(\gamma_{1}^{\prime\prime}\gamma_{4}^{\prime\prime}\right)\left(\gamma_{2}^{\prime\prime}\gamma_{3}^{\prime\prime}\right)\left(\gamma_{1}^{\prime\prime\prime}\gamma_{4}^{\prime\prime\prime}\right)\left(\gamma_{2}^{\prime\prime\prime}\gamma_{3}^{\prime\prime\prime}\right)\end{array}\right)$

provided that we correctly identify ($U$) and ($V$). The 20 letters of the
substitution ($U$) consist of the 16 letters of the substitutions ($Z$) and,
in addition, of $x_{2},x_{3},x_{17},x_{14}$, which are in four different
cycles of ($U$). Then, so that ($P$) can coincide with one of the
substitutions ($Z$) or one of their derivatives, which are all similar to
them, we will do

$y_{0}=x_{2},y_{0}^{\prime}=x_{3},y_{0}^{\prime\prime}=x_{17},y_{0}^{\prime\prime\prime}=x_{14}$

by identifying ($U$) and ($V$), and it will result

$y_{1}=x_{16},y_{2}=x_{9},y_{3}=x_{6},y_{4}=x_{8}$
$y_{1}^{\prime}=x_{12},y_{2}^{\prime}=x_{13},y_{3}^{\prime}=x_{18},y_{4}^{\prime}=x_{4}$
$y_{1}^{\prime\prime}=x_{10},x_{2}^{\prime\prime}=x_{11},y_{3}^{\prime\prime}=x_{22},y_{4}^{\prime\prime}=x_{7}$
$y_{1}^{\prime\prime\prime}=x_{10},y_{2}^{\prime\prime\prime}=x_{21},y_{3}^{\prime\prime\prime}=x_{20},y_{10}^{\prime\prime\prime}=x_{15}$

We conclude, for the substitution ($P$)

$(H)\begin{cases}(x_{16}x_{8})(x_{9}x_{6})(x_{12}x_{4})(x_{12}x_{18})\\\
(x_{10}x_{7})(x_{11}x_{22})(x_{19}x_{15})(x_{21}x_{20})\\\ \end{cases}$

which precisely represents the fourth of the substitutions ($Z$).

If we take the notation of $x^{\prime}$ and $x^{\prime}$, this substitution
could be written

$\text{(K)}\left\\{\begin{array}[]{l}\left(x_{4}^{\prime}x_{3}^{\prime}\right)\left(x_{5}^{\prime}x_{0}^{\prime}\right)\left(x_{10}^{\prime}x_{2}^{\prime}\right)\left(x_{7}^{\prime}x_{6}^{\prime}\right)\\\
\left(x_{1}^{\prime},x_{9}^{v}\right)\left(x_{4}^{\prime}x_{5}^{\prime}\right)\left(x_{7}^{\prime}x_{8}^{*}\right)\left(x_{8}^{\prime}x_{2}^{*}\right)\end{array}\right.$

Now the first cycle of $(B)$ or of $(C)$ and the first line of $(K)$ are the
two substitutions ($L$)

$\begin{array}[]{l}\left(x_{0}^{\prime}x_{1}^{\prime}x_{2}^{\prime}\ldots
x_{10}^{\prime}\right)\\\
\left(x_{9}^{\prime}x_{3}^{\prime}\right)\left(x_{\mathrm{b}}^{\prime}x_{\theta}^{\prime}\right)\left(x_{10}^{\prime}x_{t}^{\prime}\right)\left(x_{z}^{\prime}x_{\theta}^{\prime}\right)\end{array}$

which characterise a twice transitive function of 11 quantities which has
$6\times 10\times{11}$ equal values (see $\mathbf{n}^{\circ}\mathbf{5}$). The
second cycle of ($C$) and the second line of ($K$) give the two substitutions

$\text{(L')}x_{6}^{\prime}x_{1}^{n}x_{2}^{\prime\prime}\dots x_{10}^{\prime}$
$\text{(M')}(x_{1}^{\prime\prime}x_{9}^{\prime\prime})(x_{4}^{\prime\prime}x_{5}^{\prime\prime})(x_{7}^{\prime\prime}x_{8}^{\prime\prime})(x_{6}^{\prime\prime}x_{2}^{\prime\prime})$

which also characterise a twice transitive function of 11 quantities.

_So the four times transitive function of $23$ quantities remains invariable
if we make on the $11$ quantities of the first cycle of ($B$) the $6\times
10\times 1$ I substitutions derived from ($1$) and ($M$) which characterise
the twice transitive function of these $11$ quantities, provided that at the
same time we do on the quantities of the second cycle of ($B$) sub. Similar
statements similarly derived from ($L^{\prime}$) and ($M^{\prime}$), which
also characterize a twice transitive function._

_There exists a five times transitive function of $24$ quantities, which,
considered as a function of the first $23$ $x_{0},x_{1},\dots,x_{22}$, is
identical to the function we have just obtained._

Let us denote the twenty-fourth letter by $x_{\infty}$; this function is
invariant by the substitutions

$\left(z,\frac{Az+B}{Cz+D}\right)$

for which $AD-BC$ is a quadratic residue; therefore, it is characterized by
two substitutions which characterize the function precedent of $23$ quantities
and not any of the last substitutions containing $x_{\infty}$, for example by
$(z,-\frac{1}{z})$,

$(0,\infty)(1,22)(2,11)(3,15)(4,17)(5,9)$
$(6,19)(7,13)(8,20)(10,16)(12,21)(18,14)$

From what we have seen, any transitive function of $23$ quantities can be
considered invariant by substitution ($A$) and the substitution ($U$); but it
is still necessary to wonder if, besides the function which we obtain, there
does not exist another one that is invariant not only by ($z$, $5^{2}$z), but
also by the circular substitution of 22 letters ($z$, $5z$). However, it will
easily be recognized that such a function does not exist.

Now that there is only one function transitive of $23$ quantities, it is
evident that that function is characterised by substitution and by any
substitutions which leave it invariant, provided that the latter can be taken
outside of $\frac{23\times 22}{2}$ substitutions which are of the form

$(z,az+b),$

being a quadratic residue of $23$; and if we notice that the substitutions of
that function who have the fewest letters are those which have $16$, we see
that all the derivatives of a circular substitution of $23$ letters and of a
substitution of less than $16$ letters taken among these last form the $23\\!$
possible substitutions, or only half, depending on whether the second
substitution is even or odd.

## Form of the substitution which characterizes all the preceding transitive
functions.

Let us find out in what form we can write the substitution ($D$) of
$n^{\circ}1$, which is

$\text{(D)}\left(x_{z}^{\prime},x_{mz}^{\prime})(x_{z}^{\prime\prime},x_{mz}^{\prime\prime}\right),$

by asking

$\gamma^{u}\equiv{m},\quad(\bmod.\frac{p-1}{2}),$

when we restore the $x$ without accents: $x_{1},x_{2},\dots$.

The first part

$\text{(L)}\left(x_{z}^{\prime},x_{mz}^{\prime}\right)$

of the substitution (D) is carried out on the quantities
$x_{0}^{\prime},x_{1}^{\prime},x_{2}^{\prime},\dots$, or

$x_{g^{0}},x_{g^{2}},x_{g^{4}},\dots,x_{g^{p-3}},$

whose indices are quadratic residuals according to the modulus $p$; it comes
down to this substitution on the exponents of the even powers of $g$

$(2t,2mt),\quad(\bmod.p-1);$

thus $g^{2t}$ is changed to $g^{2mt}$; therefore by denoting $a$ an arbitrary
quadratic residue with respect to $p$, the substitution $(\mathrm{L})$ can be
written.

$\left(a,a^{m}\right).$

Consider the second part

$\text{(M)}\qquad\left(x_{z}^{\prime\prime},x_{m_{z}}^{\prime\prime}\right)$

substitution (D), it is performed on $x_{g},x_{g^{3}},x_{g^{5}},\dots;$ only
one of these quantities is not moved, and we represent it by $x_{g^{\beta}}$,
$\beta$ being odd. So in general, $g^{\beta+2t}$ will be changed to
$g^{\beta+2mt}$. If we represent the non-residues of $p$ through $b$ and we
ask, depending on the module $p$,

$g^{\beta+2t}\equiv b,$

we’ll have

$g^{\beta+2mt}=g^{(1-m)\beta}b^{m}$

and the substitution (M) can be written as

$\left(b,g^{(1-m)\beta}b^{m}\right)$

We have just obtained two formulas to represent the substitution ($D$),
depending on whether they are indices which are quadratic residues or indices
which are non-residual; but we can represent it by the only formula

$\left(z,\mathrm{A}z^{\frac{p-1}{2}+m}+\mathrm{B}z^{m}\right)$

In fact, if $a$ is a residue of $p$, we have $a^{\frac{p-1}{2}}\equiv
1(\bmod.p)$, and, by expressing that the substitution changes $a$ to $a^{m}$,
we get

$\mathrm{A}+\mathrm{B}\equiv 1.$

If $b$ is a non-residue of $p$, we have
$b^{\frac{p-1}{2}}\equiv{-1}(\bmod.p)$, and, in using the second part of the
substitution, we have

$-\mathrm{A}+\mathrm{B}\equiv g^{(1-m)\beta}.$

We conclude

$\mathrm{A}\equiv\frac{1-g^{\beta(1-m)}}{2},\quad\mathrm{B}\equiv\frac{1+g^{\beta(1-m)}}{2}.$

The substitutions that we have designated, in the $n^{\circ}\mathbf{2}$, by
$\mathbf{1}^{\circ},\mathbf{2}^{\circ},\mathbf{3}^{\circ}$ can be written
respectively

$\begin{array}[]{l}\left(z,-2z^{5}+3z^{2}\right)(\bmod 7),\\\
\left(z,z^{5}\right),\\\ \left(z,-z^{5}-2z^{2}\right);\end{array}$

therefore the two-time transitive function of seven letters is characterized
by the substitution $(z,z+1)$ and any of the three preceding substitutions.

The substitutions that we have designated, in the $n^{\circ}\mathbf{3}$, by
$2^{\circ}$ and $4^{\circ}$ can be written, according to modulo $11$,

$\begin{array}[]{l}\left(z,5z^{9}-4z^{4}\right),\\\
\left(z,3z^{9}-2z^{4}\right);\end{array}$

therefore the eleven letter twice transitive function is characterized by
$(z,z+1)$ and one of these two substitutions.

We then see that the four times transitive function of $11$ letters is
determined by $(z,z+1)$ and one of the two substitutions

$\begin{array}[]{c}\left(z,-3z^{7}+4z^{2}\right)\\\
\left(z,2z^{7}-z^{2}\right)\end{array}$

Finally the four times transitive function of $23$ quantities is given by
$(z,z+1)$ and by substitution

$\left(z,-3z^{15}+4z^{4}\right)\text{. }$

## Reflections on the transitive functions of a prime number of quantities.

According to the previous calculations, there is a transitive function of $7$
quantities which is twice, two transitive functions of $11$ quantities, one of
which is twice and the other four times, finally a single transitive function
of $23$ quantities which is four times. If we generally denote by $p$ the
number of quantities enclosed in these functions, these functions are
invariant by the substitutions of the form

$\text{(g)}\left(z,a^{2}z+b\right),\hfill$

and, $\infty$ being the index of a new quantity, they give functions of $p+1$
quantities which are transitive once more and invariant by the substitutions

$\left(z,\frac{\mathrm{A}z+\mathrm{B}}{\mathrm{C}z+\mathrm{D}}\right)$

for which $AD-BC$ is a quadratic residue.

Between $11$ and $23$ are the three prime numbers $13,17,$ and $19$, and I had
to wonder if there are any transitive functions of $13,17$ or $19$ letters
invariant by all the substitutions ($g$). After doing the necessary
calculations, I recognized that there were none. From the tests that I have
made, I am, moreover, extremely inclined to believe: that there are no
multiple transitive functions of $13$ or of $19$ letters. As for the
transitive functions of $17$ quantities, there are three, and which are three
times transitive, provided by this theorem that I gave in my Memoir on the
functions of several quantities (Chapter II): ”If $\tau$ is a divisor of
$\nu$, there exists a three times transitive function of $p^{\nu}+1$
quantities, which has
$\frac{1.2.3\ldots\left(p^{\nu}-2\right)\times\tau}{\nu}$ values. We get them
by making $p=2,\nu=4$ and $\tau$ successively equal to $1,2,4.$”

Thus the several times transitive functions of $7,11$ and $23$ quantities, and
those of $8,12$ and $24$, which have been calculated in the current Memoir are
indeed due to the fact that the prime numbers $7,11$ and $23$ are doubles of
numbers first increases by one unit. Moreover, as it is easy to see, a
transitive function whose prime number increases by one unit is at least twice
transitive.

## Acknowledgements

We are very grateful for the encouragement and support of the parents of our
talented young translators many of whom are not yet even in middle school.
This was done as part of our project to express $M_{23}$ as additive functions
on the finite field of $2^{11}$ elements.

We would also like to thank the help from two really useful websites. We could
never have read some of the characters without the help of this OCR. Also to
aid in translating, we have used Google Translate.

https://www.i2ocr.com/free-online-french-ocr
https://translate.google.com/?sl=fr&tl=en&op=translate


2023 005 23 January

In-jet $\Lambda_{\textnormal{c}}^{+}$ production in pp collisions at
$\sqrt{s}=13~{}\mathrm{TeV}$

ALICE Collaboration ALICE Collaboration

Recent measurements of charm-baryon production in hadronic collisions have
questioned the universality of charm-quark fragmentation across different
collision systems. In this work the fragmentation of charm quarks into charm
baryons is probed, by presenting the first measurement of the longitudinal jet
momentum fraction carried by $\Lambda_{\rm c}^{+}$ baryons,
$z^{\mathrm{ch}}_{\mathrm{||}}$, in hadronic collisions. The results are
obtained in proton–proton (pp) collisions at $\sqrt{s}=13$ TeV at the LHC,
with $\Lambda_{\rm c}^{+}$ baryons and track-based jets reconstructed in the
transverse momentum intervals of $3\leq p_{\rm T}^{\Lambda_{\rm c}^{+}}<15$
GeV/$c$ and $7\leq p_{\rm T}^{\rm jet\;ch}<15$ GeV/$c$, respectively. The
$z^{\mathrm{ch}}_{\mathrm{||}}$ distribution is compared to a measurement of
${\rm D}^{0}$-tagged charged jets in pp collisions as well as to PYTHIA 8
simulations. The data hints that the fragmentation of charm quarks into charm
baryons is softer with respect to charm mesons, as predicted by hadronisation
models which include colour correlations beyond leading-colour in the string
formation.

Heavy-flavour hadrons are produced in high-energy particle collisions through
the fragmentation of heavy (charm and beauty) quarks, which typically
originate in hard scattering processes in the early stages of the collisions.
The most common theoretical approach to describe heavy-flavour production in
hadronic collisions is based on the quantum chromodynamics (QCD) factorisation
approach [1], and consists of a convolution of three independent terms: the
parton distribution functions of the incoming hadrons, the cross sections of
the partonic scattering producing the heavy quarks, and the fragmentation
functions that parametrise the evolution of a heavy quark into given species
of heavy-flavour hadrons. As the transition of quarks to hadrons cannot be
described in perturbation theory, the fragmentation functions cannot be
calculated and must be extracted from data.

Fragmentation functions of charm quarks to charm baryons and mesons have been
constrained in $\mathrm{e^{+}e^{-}}$ and $\mathrm{e^{-}p}$ collisions [2, 3,
4, 5], using a variety of different observables, such as the hadron momentum
as a fraction of its maximum possible momentum, as dictated by the centre-of-
mass energy of the collision. Another method to probe the fragmentation of
quarks to hadrons is to parametrise the hadron momentum in relation to the
momentum of jets, which are collimated bunches of hadrons giving experimental
access to the properties of the scattered quark. Recently, the production of
charm mesons in jets, probed via the fractional longitudinal momentum of the
jet carried by the D meson, was measured in pp collisions at the Large Hadron
Collider (LHC) [6, 7, 8] and appears consistent with Monte Carlo (MC)
simulations tuned on $\mathrm{e^{+}e^{-}}$ data. These measurements support
the assumption of fragmentation universality across collision systems in the
charm-meson sector. This assumption underpins theoretical calculations
describing the production of heavy-flavour hadrons in hadronic collisions,
which make use of fragmentation functions tuned on $\mathrm{e^{+}e^{-}}$ and
$\mathrm{e^{-}p}$ data.

Measurements of the production cross sections of baryons in pp collisions have
questioned the hypothesis of fragmentation universality across collision
systems [9]. In the charm sector, which provides a clean probe of
hadronisation phenomena due to the large mass of the charm quark, recent
measurements performed by the ALICE Collaboration [10, 11, 12, 13, 14, 15, 16,
17, 18] in pp collisions have shown that the ratio of the
$\Lambda_{\textnormal{c}}^{+}$ (and other charm baryons) and $\rm D^{0}$
production cross sections measured at low $p_{\rm T}$ ($\lesssim 12$
$\mathrm{GeV}/c$) is significantly larger than the value expected from MC
simulations in which the charm fragmentation is tuned on $\mathrm{e^{+}e^{-}}$
and $\mathrm{e^{-}p}$ measurements, such as PYTHIA 8 [19] with the Monash tune
[20] or HERWIG 7 [21]. A recent measurement of the
$\Lambda_{\textnormal{c}}^{+}/\rm D^{0}$ ratio in pp collisions, performed by
the ALICE Collaboration in intervals of charged-particle multiplicity, also
points to a substantial increase of the $\Lambda_{\textnormal{c}}^{+}/\rm
D^{0}$ ratio with increasing multiplicity, with respect to
$\mathrm{e^{+}e^{-}}$ collisions, starting at very low multiplicities [14].

The study of charm-baryon production in jets can provide more differential
insights into hadronisation mechanisms in pp collisions, compared to $p_{\rm
T}$-differential cross sections and yield ratios of heavy-flavour hadrons,
allowing for a more accurate study of the dynamical properties of baryon
production. In this letter, the first measurement of the longitudinal momentum
fraction of the jet carried by $\Lambda_{\textnormal{c}}^{+}$ baryons,
$z^{\mathrm{ch}}_{\mathrm{||}}$, is presented. The measurement is performed in
pp collisions at $\sqrt{s}=13$ $\mathrm{TeV}$ in the interval $0.4\leq
z^{\mathrm{ch}}_{\mathrm{||}}\leq 1.0$. The $z^{\mathrm{ch}}_{\mathrm{||}}$
distribution, fully corrected to particle level, is presented for prompt
(charm-quark initiated) $\Lambda_{\textnormal{c}}^{+}$-tagged jets with $7\leq
p_{\rm T}^{\rm jet\;ch}<15$ $\mathrm{GeV}/c$ and $3\leq
p_{\textnormal{T}}^{\Lambda_{\textnormal{c}}^{+}}<15$ $\mathrm{GeV}/c$. The
results are then compared to PYTHIA 8 simulations [19, 22], including a
version where mechanisms beyond the leading-colour approximation are
considered in string formation processes during hadronisation [20], and to an
analogous measurement of the $z^{\mathrm{ch}}_{\mathrm{||}}$ distribution of
$\rm D^{0}$ mesons, performed by the ALICE Collaboration [6].

A full description of the ALICE setup and apparatus can be found in Refs. 23,
24. The main detectors used in this analysis are the Inner Tracking System
(ITS), which is used for vertex reconstruction and tracking; the Time
Projection Chamber (TPC), which is used for tracking and particle
identification (PID); and the Time-Of-Flight (TOF) detector, which is used for
PID. These detectors cover a pseudorapidity interval of $|\eta|<0.9$. The
analysis was performed on pp collisions at $\sqrt{s}=13~{}\mathrm{TeV}$,
collected using a minimum-bias (MB) trigger during the years 2016, 2017, and
2018. The trigger condition required coincident signals in the two
scintillator arrays of the V0 detector, with background events originating
from beam–gas interactions removed offline using timing information from the
V0. To mitigate against pile-up effects, events with multiple reconstructed
primary vertices were rejected. To ensure uniform acceptance, only events with
a primary-vertex position along the beam axis direction of $|z_{\rm vtx}|<10$
cm around the nominal interaction point were accepted. After the selections
described above, the data sample consisted of $1.7\times 10^{9}$ events,
corresponding to an integrated luminosity of
$\mathcal{L}_{\text{int}}=29~{}\mathrm{nb}^{-1}$ [25].

The $\Lambda_{\textnormal{c}}^{+}$ candidates and their charge conjugates were
reconstructed via the hadronic
$\Lambda_{\textnormal{c}}^{+}\to\textnormal{p}\textnormal{K}^{0}_{\textnormal{S}}\to\textnormal{p}\uppi^{+}\uppi^{-}$
decay channel with a total branching ratio of $(1.10\pm 0.06)\%$ [26], in the
$\Lambda_{\textnormal{c}}^{+}$ transverse-momentum interval of $3\leq
p_{\textnormal{T}}^{\Lambda_{\textnormal{c}}^{+}}<15$ $\mathrm{GeV}/c$. Only
tracks with $|\eta|<0.8$ and $p_{\rm T}>0.4$ $\mathrm{GeV}/c$, which fulfilled
the track quality selections described in Ref. 13, were considered for the
$\Lambda_{\textnormal{c}}^{+}$ reconstruction. The
$\Lambda_{\textnormal{c}}^{+}$ candidates themselves were reconstructed in the
$|y^{\Lambda_{\textnormal{c}}^{+}}|<0.8$ rapidity interval. The
$\Lambda_{\textnormal{c}}^{+}$-candidate selection was performed using a
multivariate technique based on the Boosted Decision Tree (BDT) algorithm
provided by the XGBoost package [27]. The features considered in the
optimisation include the PID signal for the proton track, the invariant mass
of the $\textnormal{K}^{0}_{\textnormal{S}}$-meson candidate, and topological
variables that exploit the kinematic properties of the displaced
$\textnormal{K}^{0}_{\textnormal{S}}$-meson decay vertex. The training was
performed in intervals of $\Lambda_{\textnormal{c}}^{+}$-candidate $p_{\rm
T}$, considering prompt signal candidates from PYTHIA 8 events with the Monash
tune [19, 20], transported through a realistic description of the detector
geometry and material budget using GEANT 3 [28]. Background candidates were
extracted from the sidebands of the invariant-mass distributions in data. The
probability thresholds of the BDT selections were tuned, using MC simulations,
to maximise the statistical significance for the signal. Further details on
the $\Lambda_{\textnormal{c}}^{+}$-candidate reconstruction and machine
learning procedure are provided in Ref. 14, where the same reconstruction and
BDT model were employed.

For the events where at least one selected $\Lambda_{\textnormal{c}}^{+}$
candidate was identified, a jet-finding procedure was performed, using the
FastJet package [29]. Prior to jet clustering, the
$\Lambda_{\textnormal{c}}^{+}$-candidate daughter tracks were replaced by the
reconstructed $\Lambda_{\textnormal{c}}^{+}$-candidate four-momentum vector.
Track-based jet finding was carried out on charged tracks with $|\eta|<0.9$
and $p_{\rm T}>0.15$ $\mathrm{GeV}/c$, using the anti-$k_{\rm T}$ algorithm
[30], with a resolution parameter of $R=0.4$. Tracks were combined using the
$E$-scheme recombination [31], with the jet transverse momentum limited to the
interval of $5\leq p_{\rm T}^{\rm jet\;ch}<35$ $\mathrm{GeV}/c$. The full jet
cone was required to be within the ALICE central barrel acceptance, limiting
the jet axis to the interval $|\eta_{\rm jet}|<0.5$. Only jets tagged via the
presence of a reconstructed $\Lambda_{\textnormal{c}}^{+}$ candidate amongst
their constituents were considered for the analysis. For events where more
than one $\Lambda_{\textnormal{c}}^{+}$ candidate was found, the jet finding
and tagging pass was performed independently for each candidate, with only the
daughters of that particular candidate replaced by the corresponding
$\Lambda_{\textnormal{c}}^{+}$ four-vector each time. In mechanisms of
hadronisation that include colour correlations beyond the leading-colour
approximation [20], which have been shown to be relevant in hadronic
collisions at LHC energies [9], hadrons can be formed in processes that
combine quarks from the parton shower with those from the underlying event
[32]. As such, the underlying event is not well defined with respect to the
measured hadron distributions. Therefore no underlying event correction is
implemented in this work.

The fragmentation of charm quarks to $\Lambda_{\textnormal{c}}^{+}$ baryons is
probed by measuring the fraction of the jet momentum carried by the
$\Lambda_{\textnormal{c}}^{+}$ along the direction of the jet axis,
$z^{\mathrm{ch}}_{\mathrm{||}}$. This is calculated for each jet using

$z^{\mathrm{ch}}_{\mathrm{||}}=\frac{\bm{p}_{\textnormal{jet}}\cdot\bm{p}_{\Lambda_{\textnormal{c}}^{+}}}{\bm{p}_{\textnormal{jet}}\cdot\bm{p}_{\textnormal{jet}}},$
(1)

where $\bm{p}_{\textnormal{jet}}$ and $\bm{p}_{\Lambda_{\textnormal{c}}^{+}}$
are the jet and $\Lambda_{\textnormal{c}}^{+}$ three-momentum vectors,
respectively.

The $z^{\mathrm{ch}}_{\mathrm{||}}$ distributions of true
$\Lambda_{\textnormal{c}}^{+}$-tagged jets were extracted in intervals of
$\Lambda_{\textnormal{c}}^{+}$ $p_{\rm T}$ and $p_{\rm T}^{\rm jet\;ch}$ using
a sideband subtraction procedure. To enact this subtraction, the invariant-
mass ($m_{\rm inv}$) distributions of $\Lambda_{\textnormal{c}}^{+}$
candidates, obtained for each $\Lambda_{\textnormal{c}}^{+}$ $p_{\rm T}$ and
$p_{\rm T}^{\rm jet\;ch}$ interval, were fitted with a function comprising a
Gaussian for the signal and an exponential for the background. The fit
parameters were then used to define signal (containing the majority of true
signal candidates) and sideband (entirely composed of background candidates)
regions, defined by $|m_{\rm inv}-\mu_{\rm fit}|<2\sigma_{\rm fit}$ and
$4\sigma_{\rm fit}<|m_{\rm inv}-\mu_{\rm fit}|<9\sigma_{\rm fit}$,
respectively, where $\mu_{\rm fit}$ and $\sigma_{\rm fit}$ represent the mean
and sigma of the fitted Gaussian distributions. The
$z^{\mathrm{ch}}_{\mathrm{||}}$
($p_{\textnormal{T}}^{\Lambda_{\textnormal{c}}^{+}}$,$p_{\rm T}^{\rm
jet\;ch}$) distributions were extracted in the signal and sideband regions,
with the sideband distribution scaled by the ratio of the background function
integrals in the signal and sideband regions. The sideband distribution was
then subtracted from the signal one, with the resulting distribution scaled to
account for the fact that the $2\sigma_{\rm fit}$ width of the signal region
only encompasses approximately $95\%$ of the total signal, to obtain the
sideband subtracted $z^{\mathrm{ch}}_{\mathrm{||}}$ yield in each
$p_{\textnormal{T}}^{\Lambda_{\textnormal{c}}^{+}}$ and $p_{\rm T}^{\rm
jet\;ch}$ interval.

To account for the reconstruction and selection efficiency of the
$\Lambda_{\textnormal{c}}^{+}$-tagged jet signal, the sideband subtracted
$z^{\mathrm{ch}}_{\mathrm{||}}$ distributions in each
$p_{\textnormal{T}}^{\Lambda_{\textnormal{c}}^{+}}$ and $p_{\rm T}^{\rm
jet\;ch}$ interval,
$N(z^{\mathrm{ch}}_{\mathrm{||}},p_{\textnormal{T}}^{\Lambda_{\textnormal{c}}^{+}},p_{\rm
T}^{\rm jet\;ch})$, were scaled by the reconstruction efficiency of prompt
$\Lambda_{\textnormal{c}}^{+}$-tagged jets, $\epsilon_{\textnormal{prompt}}$,
and summed over the entire $p_{\textnormal{T}}^{\Lambda_{\textnormal{c}}^{+}}$
interval to obtain the efficiency-corrected $z^{\mathrm{ch}}_{\mathrm{||}}$
yield of $\Lambda_{\textnormal{c}}^{+}$-tagged jets,
$N^{\textnormal{corr}}(z^{\mathrm{ch}}_{\mathrm{||}},p_{\rm T}^{\rm
jet\;ch})$, given by

$N^{\text{corr}}(z^{\mathrm{ch}}_{\mathrm{||}},p_{\rm T}^{\rm
jet\;ch})=\sum_{p_{\textnormal{T}}^{\Lambda_{\textnormal{c}}^{+}}}\frac{N(z^{\mathrm{ch}}_{\mathrm{||}},p_{\textnormal{T}}^{\Lambda_{\textnormal{c}}^{+}},p_{\rm
T}^{\rm
jet\;ch})}{\epsilon_{\rm{prompt}}(p_{\textnormal{T}}^{\Lambda_{\textnormal{c}}^{+}})}.$
(2)

The
$\epsilon_{\textnormal{prompt}}(p_{\textnormal{T}}^{\Lambda_{\textnormal{c}}^{+}})$
efficiency is strongly dependent on
$p_{\textnormal{T}}^{\Lambda_{\textnormal{c}}^{+}}$, ranging from about $20\%$
at $3<p_{\textnormal{T}}^{\Lambda_{\textnormal{c}}^{+}}<4~{}\mathrm{GeV}/c$ to
$40\%$ at
$12<p_{\textnormal{T}}^{\Lambda_{\textnormal{c}}^{+}}<24~{}\mathrm{GeV}/c$,
and was calculated using PYTHIA 8 simulations with the Monash tune containing
prompt $\Lambda_{\textnormal{c}}^{+}$-tagged jets, transported through the
detector using GEANT 3. This efficiency does not exhibit a $p_{\rm T}^{\rm
jet\;ch}$ dependence.

In order to isolate the $N^{\text{corr}}(z^{\mathrm{ch}}_{\mathrm{||}},p_{\rm
T}^{\rm jet\;ch})$ distribution of prompt
$\Lambda_{\textnormal{c}}^{+}$-tagged jets, a feed-down subtraction was
employed to remove the non-prompt (beauty-quark initiated) contribution. The
non-prompt cross section was obtained from particle level POWHEG [33] \+
PYTHIA 6 [34] \+ EvtGen [35] simulations, as a function of $p_{\rm T}^{\rm
jet\;ch}$, $p_{\textnormal{T}}^{\Lambda_{\textnormal{c}}^{+}}$ and
$z^{\mathrm{ch}}_{\mathrm{||}}$, and was scaled according to the integrated
luminosity of the analysed data sample and the branching ratio of the
$\Lambda_{\textnormal{c}}^{+}\to\textnormal{p}\textnormal{K}^{0}_{\textnormal{S}}\to\textnormal{p}\uppi^{+}\uppi^{-}$
decay channel. The resulting particle-level yield was multiplied by the ratio
of the non-prompt to prompt $\Lambda_{\textnormal{c}}^{+}$-tagged jet
reconstruction and selection efficiency in intervals of
$p_{\textnormal{T}}^{\Lambda_{\textnormal{c}}^{+}}$ and integrated over the
$p_{\textnormal{T}}^{\Lambda_{\textnormal{c}}^{+}}$ range. The simulated non-
prompt results were then folded to reconstructed level, using a four-
dimensional response matrix generated using non-prompt
$\Lambda_{\textnormal{c}}^{+}$-tagged jets in PYTHIA 8 with the Monash tune,
transported through a simulation of the ALICE detector using GEANT 3. The
response matrix was constructed as a function of $p_{\rm T}^{\rm jet\;ch}$ and
$z^{\mathrm{ch}}_{\mathrm{||}}$ at generator and reconstruction levels. The
folded results were then subtracted from the measured
$N^{\text{corr}}(z^{\mathrm{ch}}_{\mathrm{||}},p_{\rm T}^{\rm jet\;ch})$
distribution in data, removing the non-prompt contribution. The estimated
fraction of $\Lambda_{\textnormal{c}}^{+}$-tagged jets coming from b-quark
fragmentation is found to be about 5$\%$, with no significant
$z^{\mathrm{ch}}_{\mathrm{||}}$ dependence.

A two-dimensional Bayesian unfolding procedure [36] was performed to correct
for detector effects and obtain the $z^{\mathrm{ch}}_{\mathrm{||}}$
distribution for prompt $\Lambda_{\textnormal{c}}^{+}$-tagged jets at particle
level. A four-dimensional response matrix as a function of $p_{\rm T}^{\rm
jet\;ch}$ and $z^{\mathrm{ch}}_{\mathrm{||}}$, at generator and reconstruction
levels, was populated with prompt $\Lambda_{\textnormal{c}}^{+}$-tagged jets,
obtained with PYTHIA 8 simulations with the Monash tune, passed through a
simulation of the ALICE detector using GEANT 3. The measured data and response
matrix were provided in the intervals of $5\leq p_{\rm T}^{\rm jet\;ch}<35$
$\mathrm{GeV}/c$ and $0.4\leq z^{\mathrm{ch}}_{\mathrm{||}}\leq 1.0$, with the
final unfolded results reported in the intervals $7\leq p_{\rm T}^{\rm
jet\;ch}<15$ $\mathrm{GeV}/c$ and $0.4\leq z^{\mathrm{ch}}_{\mathrm{||}}\leq
1.0$. Corrections accounting for migrating entries in and out of the response
matrix ranges, as modelled by the same MC simulation, were also applied. The
corrected $z^{\mathrm{ch}}_{\mathrm{||}}$ distribution is normalised to the
total number of $\Lambda_{\textnormal{c}}^{+}$-tagged jets in the reported
$z^{\mathrm{ch}}_{\mathrm{||}}$ and $p_{\rm T}^{\rm jet\;ch}$ interval.

The systematic uncertainties affecting the measurement were evaluated, in each
$z^{\mathrm{ch}}_{\mathrm{||}}$ interval, by modifying the strategy adopted at
various steps of the analysis procedure and assessing the impact on the
unfolded $z^{\mathrm{ch}}_{\mathrm{||}}$ distribution. The total systematic
uncertainty includes contributions from multiple sources. The first considered
source is the sideband subtraction procedure (ranging from 3.7% to 7.6%
depending on the $z^{\mathrm{ch}}_{\mathrm{||}}$ inteval), whose contribution
was estimated by varying the invariant-mass fit parameters as well as the
invariant-mass intervals of the signal and sideband regions. The contribution
from the BDT selection of $\Lambda_{\textnormal{c}}^{+}$ candidates (from 7.3%
to 19%) was estimated by varying the BDT probability thresholds to induce a
$25\%$ variation in the $\Lambda_{\textnormal{c}}^{+}$-tagged jet
reconstruction and selection efficiency. The uncertainty from the jet energy
resolution (from 4.5% to 19%) was estimated by recalculating the response
matrix used for unfolding with a $4\%$ reduced tracking efficiency. The
reduction in the tracking efficiency was evaluated by varying the track-
selection criteria and propagating the ITS–TPC track-matching efficiency
uncertainty. The uncertainty on the feed-down subtraction ($<2\%$) was
estimated by varying the choice of POWHEG parameters considered to generate
the feed-down cross section, including the factorisation and renormalisation
scales, as well as the mass of the beauty quark, which were varied according
to theoretical prescriptions [37]. Finally the contribution from the unfolding
procedure (from 1.1% to 2.7%) was estimated by altering the choice of prior,
regularisation parameter, and ranges of the response matrix. For each of the
aforementioned categories, several variations were made and the root-mean-
square of the resulting distributions was considered. The exceptions are
related to the contribution associated to the choice of parameters of the
POWHEG calculations, where only the largest deviation from the central result,
in each direction, was considered, as well as the uncertainty on the jet
energy resolution where the variation with respect to the central result was
taken as the uncertainty. All uncertainties (other than from the feed-down
subtraction) were then symmetrised. The uncertainties were combined in
quadrature to obtain the total systematic uncertainty on the measurement,
which ranges from $13\%$ to $28\%$.

Figure 1: (Left) Fully corrected $z^{\mathrm{ch}}_{\mathrm{||}}$ distribution
of $\Lambda_{\textnormal{c}}^{+}$-tagged track-based jets (black open circles)
measured in the $7\leq p_{\rm T}^{\rm jet\;ch}<15$ $\mathrm{GeV}/c$ and $3\leq
p_{\textnormal{T}}^{\Lambda_{\textnormal{c}}^{+}}<15$ $\mathrm{GeV}/c$
intervals in pp collisions at $\sqrt{s}=13$ TeV, compared with predictions
from different PYTHIA 8 tunes [19, 20, 22] (red-dotted and green-dashed
lines). The ratios of the MC simulations to the data are shown in the bottom
panel. (Right) Comparison of the measured $z^{\mathrm{ch}}_{\mathrm{||}}$
distribution of $\Lambda_{\textnormal{c}}^{+}$-tagged jets and the previously
measured $z^{\mathrm{ch}}_{\mathrm{||}}$ distribution of $\rm D^{0}$-tagged
jets [6], obtained in the same kinematic interval. The ratio of the
$z^{\mathrm{ch}}_{\mathrm{||}}$ distribution of
$\Lambda_{\textnormal{c}}^{+}$-tagged and $\rm D^{0}$-tagged jets is shown in
the bottom panel for both the data and the different PYTHIA tunes.

The fully corrected $z^{\mathrm{ch}}_{\mathrm{||}}$ distribution of prompt
$\Lambda_{\textnormal{c}}^{+}$-tagged track-based jets in the intervals of
$7\leq p_{\rm T}^{\rm jet\;ch}<15$ $\mathrm{GeV}/c$ and $3\leq
p_{\textnormal{T}}^{\Lambda_{\textnormal{c}}^{+}}<15$ $\mathrm{GeV}/c$ is
presented in the left-hand panel of Fig. 1 and compared to PYTHIA 8
simulations with two different tunes. In PYTHIA 8 the Lund string model of
fragmentation is employed, where endpoints are confined by linear potentials
encoded in strings. For the case of heavy quarks, the Lund fragmentation
function is modified to account for the slower propagation of the massive
endpoints compared to their massless counterparts. The Monash tune (red-dotted
line) [19], in which the charm fragmentation is tuned on $\mathrm{e^{+}e^{-}}$
measurements, predicts a harder fragmentation than the measurement. An
evaluation of the $\chi^{2}/$ndf between the measured data points and the
model was performed, combining the statistical and systematic uncertainties on
the data in quadrature and assuming the uncertainties are uncorrelated across
the $z^{\mathrm{ch}}_{\mathrm{||}}$ intervals. This exercise determines that
there is a $0.4\%$ probability that the model describes the data. A better
agreement is achieved by the PYTHIA 8 with the CR-BLC Mode 2 tune, that
includes colour reconnection mechanisms beyond the leading-colour
approximation [22] (green-dashed line). In this model, the minimisation of the
string potential is implemented considering the SU(3) multiplet structure of
QCD in a more realistic way than in the leading-colour approximation, allowing
for the formation of “baryonic” configurations where for example two colours
can combine coherently to form an anti-colour. The same $\chi^{2}/$ndf
approach results in a $78\%$ probability that the model describes the data.
The simulation with PYTHIA 8 with the CR-BLC Mode 2 tune also provides a much
more accurate description of the $\Lambda_{\textnormal{c}}^{+}/\rm D^{0}$
cross section ratio, previously measured in pp collisions at the LHC [10, 11,
13, 12, 14, 38].

In the right-hand panel of Fig. 1, a comparison of the
$z^{\mathrm{ch}}_{\mathrm{||}}$ distribution of
$\Lambda_{\textnormal{c}}^{+}$-tagged jets and the
$z^{\mathrm{ch}}_{\mathrm{||}}$ distribution previously measured for $\rm
D^{0}$-tagged jets [6] is presented. The latter is consistent with PYTHIA 8
simulations using both the Monash and CR-BLC Mode 2 tunes. The ratio of the
two distributions is also presented in the bottom panel. The uncertainty from
the jet energy resolution was considered to be correlated between the
$\Lambda_{\textnormal{c}}^{+}$-tagged jet and $\rm D^{0}$-tagged jet
measurements and was evaluated directly on the ratio of the distributions. The
remaining uncertainties were considered uncorrelated when taking the ratio and
were then combined in quadrature with the uncertainty of the jet energy
resolution. The uncertainties were considered uncorrelated across the
$z^{\mathrm{ch}}_{\mathrm{||}}$ intervals. The same $\chi^{2}/$ndf exercise
described above determines that there is a $12\%$ probability that the
measured ratio is described by a flat distribution at unity, hinting at a
softer fragmentation of charm quarks into charm baryons than charm mesons. The
ratio is better described by the PYTHIA 8 simulations with the CR-BLC Mode 2
compared to the ones with the Monash tune, with the former describing the data
with $88\%$ probability compared to a $0.03\%$ probability for the latter.

In summary the first measurement in hadronic collisions of the longitudinal
momentum fraction of the jet carried by $\Lambda_{\textnormal{c}}^{+}$ baryons
was presented for pp collisions at $\sqrt{s}=13$ TeV. The result is fully
corrected to particle level and obtained in the jet and
$\Lambda_{\textnormal{c}}^{+}$ transverse-momentum intervals of $7\leq p_{\rm
T}^{\rm jet\;ch}<15$ $\mathrm{GeV}/c$ and $3\leq
p_{\textnormal{T}}^{\Lambda_{\textnormal{c}}^{+}}<15$ $\mathrm{GeV}/c$,
respectively. The measurement presented in this Letter hints that charm quarks
have a softer fragmentation into $\Lambda_{\textnormal{c}}^{+}$ baryons
compared to $\rm D^{0}$ mesons, in the measured kinematic interval. One
possible explanation is that charm-baryon production is favoured in the
presence of higher particle multiplicity originating from both the jet
fragmentation and the underlying event, which could be tested with future
measurements of the in-jet multiplicity of
$\Lambda_{\textnormal{c}}^{+}$-tagged jets. The fragmentation of charm quarks
into $\Lambda_{\textnormal{c}}^{+}$ baryons in hadronic collisions exhibits
tension with simulations tuned on $\mathrm{e^{+}e^{-}}$ data that employ a
leading-colour formalism of hadronisation, such as in the Monash tune of
PYTHIA 8. This occurs despite their successful description of the
fragmentation of charm quarks into $\rm D^{0}$ mesons. However, the inclusion
of mechanisms sensitive to the surrounding partonic density that feature
colour reconnection beyond the leading-colour approximation results in a
better agreement with data. This result also partially explains the $p_{\rm
T}$ shape of the prompt $\Lambda_{\textnormal{c}}^{+}/\rm D^{0}$ cross section
ratio [10, 11, 13, 12, 14, 38], which shows a peak at low $p_{\rm T}$
($\approx 3$ $\mathrm{GeV}/c$) and is also described within uncertainties by
PYTHIA 8 with the CR-BLC Mode 2 tune. The $p_{\rm T}$ trend of this ratio is
driven by the fact that the $\Lambda_{\textnormal{c}}^{+}$ baryons produced
from the fragmenting charm quark carry a significantly lower fraction of the
charm-quark transverse momentum than the $\rm D^{0}$ mesons produced in a
similar way.

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