The Foata correspondence, cycle lengths and anomalies

TL;DR

This paper demonstrates that the Foata correspondence serves as a bijection between two classes of permutations with equal cardinalities, addressing a question raised in the context of jammed configurations in theater models.

Contribution

It establishes a new connection between permutation classes and the Foata correspondence, providing a bijective proof for their equal cardinalities.

Findings

Foata correspondence acts as a bijection between the two permutation classes

Addresses a question from the study of jammed configurations in theater models

Provides a combinatorial interpretation of permutation class equivalences

Abstract

In their study of the densest jammed configurations for theater models, Krapivsky and Luck observe that two classes of permutations have the same cardinalities and ask for a bijection between them. In this note we show that the Foata correspondence provides the desired bijection.

Equations

$$X_i^s:=\left\{j\le i|a_j>\max (a_{i+1},\dots ,a_{i+b})\right\}$$

Full text

Krapivsky and Luck (2019) introduced the theater model as a variant of directed random sequential adsorption, where spectators sequentially select a seat in a row of $L$ seats, with the constraint that they cannot go past a cluster of $b$ or more consecutive occupied seats. Configurations where all the seats are eventually occupied are induced by permutations $s$ of $\{1,\ldots,L\}$ such that for any $i$ between $1$ and $L$, one cannot find $b$ consecutive integers $j+1,\ldots,j+b$ with $j+b<i$ and $s(j+k)>s(i)$ for all $k$ between $1$ and $b$. Krapivsky and Luck (2019) showed that the number $D_{L}^{(b)}$ of such permutations satisfies a linear recurrence relation which implies that they have the same cardinality as the permutations of $L$ elements with cycles of lengths at most $b$. The authors then asked for a bijective proof of this fact. The goal of this note is to show that the Foata correspondence (Foata (1968); Lothaire (1983)) provides such a bijection. Interestingly enough, the Foata correspondence is already visible in Rényi (1962), where it is used to explain the equality of the distributions of records and cycle lengths for permutations. The present note broadens the connection between generalized notions of records and cycle lengths.

In Section 1 we recall the Foata correspondence and in Section 2 we show that it provides the desired bijection.

1 The Foata correspondence

Let $\mathcal{S}_{L}$ be the group of permutations of $\{1,\ldots,L\}$. We will represent permutations in $\mathcal{S}_{L}$ by words with $L$ distinct letters in $\{1,\ldots,L\}$. Our running example will be $359724681$, which denotes the permutation $s\in\mathcal{S}_{9}$ such that $s(1)=3$, $s(2)=5$, $s(3)=9$, etc. The point diagram of a permutation is a plot of the graph of the corresponding function from $\{1,\ldots,L\}$ to itself, see Figure 1 for the point diagram of the above example.

One can associate to every permutation in $\mathcal{S}_{L}$ its cycle decomposition. Including the fixed points in that decomposition, the above $s\in\mathcal{S}_{9}$ has cycle decomposition $[139][25][476][8]$. This way of writing is however not unique for two reasons:

• •

each cycle of length $d$ can be written in $d$ different ways (one can freely choose what element to put first) ;

• •

if a permutation has $k$ cycles (including singletons corresponding to fixed points) one can have them appear in $k!$ different orders.

The Foata correspondence (Foata (1968); Lothaire (1983)) describes a canonical choice for writing such a cycle decomposition. Firstly we write every cycle by starting by its maximal element. We call the maximal element of a cycle the cycle head. In the above example, $[139]$ is written as $[913]$, $[25]$ is written as $[52]$ and $[476]$ is written as $[764]$. Secondly, we write the cycles in increasing order of their cycle heads. In the above example, we obtain $[52][764][8][913]$. Removing the brackets, we obtain the word $527648913$ which can be seen as a permutation. The Foata correspondence associates to any permutation $s\in\mathcal{S}_{L}$ the permutation $F(s)$ obtained by writing the cycle decomposition of $s$ in the above way and removing the brackets.

Remark 1.1*.*

The largest letter to the left (or at the position) of a given letter $a$ in $F(s)$ corresponds to the cycle head of the cycle to which $a$ belongs in the cycle decomposition of $s$. This observation will be used later.

2 Cycle lengths and $b$-anomalies

Definition 2.1**.**

Let $b\geq 1$ be an integer. Let $a_{1}\cdots a_{L}$ denote a permutation in $\mathcal{S}_{L}$, with $L\geq 1$. A consecutive subword $a_{i+1}\cdots a_{i+b}$ is called a $b$-anomaly if there exists $1\leq j\leq i$ such that $a_{j}>\max(a_{i+1},\ldots,a_{i+b})$.

In terms of the point diagram, a $b$-anomaly corresponds to $b$ points with consecutive abscissae for which one can find a point strictly above and to the left of all the $b$ points. In the example of Figure 1, $246$ is a $3$-anomaly because $9$ is to its left and greater than $2$, $4$ and $6$.

The following result relates the cycle lengths of a permutation $s$ to the $b$-anomalies of its image $F(s)$ under the Foata correspondence.

Proposition 2.2**.**

Let $b\geq 1$, $L\geq 1$ and $s\in\mathcal{S}_{L}$. Then $s$ has a cycle of length at least $b+1$ if and only if $F(s)$ has a $b$-anomaly.

Proof.

Assume $s$ has a cycle of length $d\geq b+1$. We write it $[c_{1}c_{2}\cdots c_{d}]$ with $c_{1}$ being the cycle head, that is, the largest element of the cycle. Then the subword $c_{2}\cdots c_{d}$ of $F(s)$ forms a $(d-1)$-anomaly. Any consecutive subword of length $b$ of this $(d-1)$-anomaly provides a $b$-anomaly for $F(s)$.

Conversely, assume $F(s)=a_{1}\cdots a_{L}$ has a $b$-anomaly $a_{i+1}\cdots a_{i+b}$. By definition of the $b$-anomaly, the set

[TABLE]

is non-empty, so $\max_{j\in X_{i}^{s}}{a_{j}}$ is well-defined and equal to some $a_{h}$. Then by Remark 1.1, in the cycle decomposition of $s$, $a_{h}$ is the head of the cycle to which each $a_{i+k}$ with $1\leq k\leq b$ belongs, so there are at least $b+1$ elements in that cycle in $s$. ∎

As a consequence of Proposition 2.2, in order to obtain the bijection requested by Krapivsky and Luck (2019), it suffices to compose the Foata correspondence with the involution sending every permutation $s\in\mathcal{S}_{L}$ to $\widetilde{s}\in\mathcal{S}_{L}$ defined by $\widetilde{s}(i)=L+1-s(L+1-i)$ for every $i$, whereby the point diagram is rotated by $180$ degrees.

Acknowledgements

The author thanks Jean-Marc Luck for introducing him to the theater model, the Institut de Physique Théorique for the hospitality during several visits and the Fondation Sciences Mathématiques de Paris for the support.