Number of cycles in the graph of $312$-avoiding permutations

Richard Ehrenborg,Sergey Kitaev,Einar Steingrımsson

doi:10.46298/dmtcs.2378

Abstract

The graph of overlapping permutations is defined in a way analogous to the De Bruijn graph on strings of symbols. However, instead of requiring the tail of one permutation to equal the head of another for them to be connected by an edge, we require that the head and tail in question have their letters appear in the same order of size. We give a formula for the number of cycles of length $d$ in the subgraph of overlapping $312$-avoiding permutations. Using this we also give a refinement of the enumeration of $312$-avoiding affine permutations. Le graphique de permutations qui se chevauchent est définie d’une manière analogue à celle du graphe de De Bruijn sur des chaînes de symboles. Cependant, au lieu d’exiger la queue d’une permutation d’égaler la tête d’un autre pour qu’ils soient reliés par un bord, nous avons besoin que la tête et la queue en question ont leurs lettres apparaissent dans le même ordre de grandeur. Nous donnons une formule pour le nombre de cycles de longueur$d$ dans le sous- graphe de chevauchement $312$-évitant permutations. L’utilisation de ce nous donnent également un raffinement de l’énumération de$312$-évitant permutations affines.

Highlights

Introduction and preliminariesOne of the classical objects in combinatorics is the De Bruijn graph
The De Bruijn graph has been much studied, especially in connection with combinatorics on words, and one of its well known properties is the fact that its number of directed cycles of length d, for d ≤ n, is given by 1 μ (d/e) qe, d e|d
A natural variation on the De Bruijn graphs is obtained by replacing words over an alphabet by permutations of the set of integers {1, 2, . . . , n}, where the overlapping condition determining directed edges in De Bruijn graphs is replaced by the condition that the head and tail of two permutations have the same standardization, that is, that their letters appear in the same order of size

Summary

Introduction and preliminaries

One of the classical objects in combinatorics is the De Bruijn graph. This is the directed graph on vertex set {0, 1, . . . , q − 1}n, the set of all strings of length n over an alphabet of size q, whose directed edges go from each vertex x1 · · · xn to each vertex x2 · · · xn+1. The De Bruijn graph has been much studied, especially in connection with combinatorics on words, and one of its well known properties is the fact that its number of directed cycles of length d, for d ≤ n, is given by 1 μ (d/e) qe, d e|d (1.1). N}, where the overlapping condition determining directed edges in De Bruijn graphs is replaced by the condition that the head and tail of two permutations have the same standardization, that is, that their letters appear in the same order of size. The connection between cycles in the graph of overlapping permutations and affine permutations goes through bi-infinite sequences.

Compositions and cyclic compositions

The graph of 312-avoiding permutations

The bijection

The inverse bijection