scholarly journals Operators of equivalent sorting power and related Wilf-equivalences

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Michael Albert ◽  
Mathilde Bouvel
Keyword(s):  
Permutation Statistics ◽  
Stack Sorting ◽  
Set Of Permutations ◽  
International Audience

International audience We study sorting operators $\textrm{A}$ on permutations that are obtained composing Knuth's stack sorting operator \textrmS and the reverse operator $\textrm{R}$, as many times as desired. For any such operator $\textrm{A}$, we provide a bijection between the set of permutations sorted by $\textrm{S} \circ \textrm{A}$ and the set of those sorted by $\textrm{S} \circ \textrm{R} \circ \textrm{A}$, proving that these sets are enumerated by the same sequence, but also that many classical permutation statistics are equidistributed across these two sets. The description of this family of bijections is based on an apparently novel bijection between the set of permutations avoiding the pattern $231$ and the set of those avoiding $132$ which preserves many permutation statistics. We also present other properties of this bijection, in particular for finding families of Wilf-equivalent permutation classes.

scholarly journals Cycles and sorting index for matchings and restricted permutations

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Svetlana Poznanović
Keyword(s):  
Permutation Statistics ◽  
Minimal Elements ◽  
Restricted Permutations ◽  
Dyck Paths ◽  
Set Of Permutations ◽  
Ferrers Board ◽  
International Audience ◽  
The Right

International audience We prove that the Mahonian-Stirling pairs of permutation statistics $(sor, cyc)$ and $(∈v , \mathrm{rlmin})$ are equidistributed on the set of permutations that correspond to arrangements of $n$ non-atacking rooks on a fixed Ferrers board with $n$ rows and $n$ columns. The proofs are combinatorial and use bijections between matchings and Dyck paths and a new statistic, sorting index for matchings, that we define. We also prove a refinement of this equidistribution result which describes the minimal elements in the permutation cycles and the right-to-left minimum letters.

scholarly journals Operators of Equivalent Sorting Power and Related Wilf-equivalences

10.37236/4119 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Michael Albert ◽  
Mathilde Bouvel
Keyword(s):  
Permutation Statistics ◽  
Stack Sorting ◽  
Set Of Permutations

We study sorting operators $\mathbf{A}$ on permutations that are obtained composing Knuth's stack sorting operator $\mathbf{S}$ and the reversal operator $\mathbf{R}$, as many times as desired. For any such operator $\mathbf{A}$, we provide a size-preserving bijection between the set of permutations sorted by $\mathbf{S} \circ \mathbf{A}$ and the set of those sorted by $\mathbf{S} \circ \mathbf{R} \circ \mathbf{A}$, proving that these sets are enumerated by the same sequence, but also that many classical permutation statistics are equidistributed across these two sets. The description of this family of bijections is based on a bijection between the set of permutations avoiding the pattern $231$ and the set of those avoiding $132$ which preserves many permutation statistics. We also present other properties of this bijection, in particular for finding pairs of Wilf-equivalent permutation classes.

scholarly journals Mixed volumes of hypersimplices

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Gaku Liu
Keyword(s):  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Type B ◽  
Combinatorial Interpretation ◽  
Mixed Volumes ◽  
Eulerian Numbers ◽  
Eulerian Number ◽  
Nous Montrons ◽  
Set Of Permutations ◽  
International Audience

International audience In this extended abstract we consider mixed volumes of combinations of hypersimplices. These numbers, called mixed Eulerian numbers, were first considered by A. Postnikov and were shown to satisfy many properties related to Eulerian numbers, Catalan numbers, binomial coefficients, etc. We give a general combinatorial interpretation for mixed Eulerian numbers and prove the above properties combinatorially. In particular, we show that each mixed Eulerian number enumerates a certain set of permutations in $S_n$. We also prove several new properties of mixed Eulerian numbers using our methods. Finally, we consider a type $B$ analogue of mixed Eulerian numbers and give an analogous combinatorial interpretation for these numbers. Dans ce résumé étendu nous considérons les volumes mixtes de combinaisons d’hyper-simplexes. Ces nombres, appelés les nombres Eulériens mixtes, ont été pour la première fois étudiés par A. Postnikov, et il a été montré qu’ils satisfont à de nombreuses propriétés reliées aux nombres Eulériens, au nombres de Catalan, aux coefficients binomiaux, etc. Nous donnons une interprétation combinatoire générale des nombres Eulériens mixtes, et nous prouvons combinatoirement les propriétés mentionnées ci-dessus. En particulier, nous montrons que chaque nombre Eulérien mixte compte les éléments d’un certain sous-ensemble de l’ensemble des permutations $S_n$. Nous établissons également plusieurs nouvelles propriétés des nombres Eulériens mixtes grâce à notre méthode. Pour finir, nous introduisons une généralisation en type $B$ des nombres Eulériens mixtes, et nous en donnons une interprétation combinatoire analogue.

scholarly journals Permutations Containing and Avoiding $\textit{123}$ and $\textit{132}$ Patterns

1999 ◽  
Vol Vol. 3 no. 4 ◽  
Author(s):  
Aaron Robertson
Keyword(s):  
Set Of Permutations ◽  
International Audience

International audience We prove that the number of permutations which avoid 132-patterns and have exactly one 123-pattern, equals $(n-2)2^{n-3}$, for $n \ge 3$. We then give a bijection onto the set of permutations which avoid 123-patterns and have exactly one 132-pattern. Finally, we show that the number of permutations which contain exactly one 123-pattern and exactly one 132-pattern is $(n-3)(n-4)2^{n-5}$, for $n \ge 5$.

scholarly journals Recursions and divisibility properties for combinatorial Macdonald polynomials

2011 ◽  
Vol Vol. 13 no. 1 (Combinatorics) ◽  
Author(s):  
Nicholas A. Loehr ◽  
Elizabeth Niese
Keyword(s):  
Hilbert Series ◽  
Permutation Statistics ◽  
Macdonald Polynomial ◽  
Integer Partition ◽  
Theoretical Methods ◽  
Major Index ◽  
International Audience ◽  
Definition Of ◽  
Fermionic Formulas ◽  
Divisibility Properties

Combinatorics International audience For each integer partition mu, let e (F) over tilde (mu)(q; t) be the coefficient of x(1) ... x(n) in the modified Macdonald polynomial (H) over tilde (mu). The polynomial (F) over tilde (mu)(q; t) can be regarded as the Hilbert series of a certain doubly-graded S(n)-module M(mu), or as a q, t-analogue of n! based on permutation statistics inv(mu) and maj(mu) that generalize the classical inversion and major index statistics. This paper uses the combinatorial definition of (F) over tilde (mu) to prove some recursions characterizing these polynomials, and other related ones, when mu is a two-column shape. Our result provides a complement to recent work of Garsia and Haglund, who proved a different recursion for two-column shapes by representation-theoretical methods. For all mu, we show that e (F) over tilde (mu)(q, t) is divisible by certain q-factorials and t-factorials depending on mu. We use our recursion and related tools to explain some of these factors bijectively. Finally, we present fermionic formulas that express e (F) over tilde ((2n)) (q, t) as a sum of q, t-analogues of n!2(n) indexed by perfect matchings.

scholarly journals Generating Functions for Permutations which Contain a Given Descent Set

10.37236/299 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Jeffrey Remmel ◽  
Manda Riehl
Keyword(s):  
Symmetric Group ◽  
Generating Functions ◽  
Symmetric Function ◽  
Schur Functions ◽  
Permutation Statistics ◽  
Finite Set ◽  
Set Of Permutations ◽  
Descent Set ◽  
Positive Integers

A large number of generating functions for permutation statistics can be obtained by applying homomorphisms to simple symmetric function identities. In particular, a large number of generating functions involving the number of descents of a permutation $\sigma$, $des(\sigma)$, arise in this way. For any given finite set $S$ of positive integers, we develop a method to produce similar generating functions for the set of permutations of the symmetric group $S_n$ whose descent set contains $S$. Our method will be to apply certain homomorphisms to symmetric function identities involving ribbon Schur functions.

scholarly journals Universal cycles for permutation classes

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Michael Albert ◽  
Julian West
Keyword(s):  
General Result ◽  
Universal Cycles ◽  
Set Of Permutations ◽  
International Audience

International audience We define a universal cycle for a class of $n$-permutations as a cyclic word in which each element of the class occurs exactly once as an $n$-factor. We give a general result for cyclically closed classes, and then survey the situation when the class is defined as the avoidance class of a set of permutations of length $3$, or of a set of permutations of mixed lengths $3$ and $4$. Nous définissons un cycle universel pour une classe de $n$-permutations comme un mot cyclique dans lequel chaque élément de la classe apparaît une unique fois comme $n$-facteur. Nous donnons un résultat général pour les classes cycliquement closes, et détaillons la situation où la classe de permutations est définie par motifs exclus, avec des motifs de taille $3$, ou bien à la fois des motifs de taille $3$ et de taille $4$.

scholarly journals Classification of bijections between 321- and 132-avoiding permutations

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Anders Claesson ◽  
Sergey Kitaev
Keyword(s):  
Permutation Statistics ◽  
Large Set ◽  
Systematic Analysis ◽  
Comprehensive Survey ◽  
International Audience

International audience It is well-known, and was first established by Knuth in 1969, that the number of 321-avoiding permutations is equal to that of 132-avoiding permutations. In the literature one can find many subsequent bijective proofs confirming this fact. It turns out that some of the published bijections can easily be obtained from others. In this paper we describe all bijections we were able to find in the literature and we show how they are related to each other (via "trivial'' bijections). Thus, we give a comprehensive survey and a systematic analysis of these bijections. We also analyze how many permutation statistics (from a fixed, but large, set of statistics) each of the known bijections preserves, obtaining substantial extensions of known results. We also give a recursive description of the algorithmic bijection given by Richards in 1988 (combined with a bijection by Knuth from 1969). This bijection is equivalent to the celebrated bijection of Simion and Schmidt (1985), as well as to the bijection given by Krattenthaler in 2001, and it respects 11 statistics (the largest number of statistics any of the bijections respect).

scholarly journals Some simple varieties of trees arising in permutation analysis

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Mathilde Bouvel ◽  
Marni Mishna ◽  
Cyril Nicaud
Keyword(s):  
Related Species ◽  
Asymptotic Formulas ◽  
Genome Comparison ◽  
Set Of Permutations ◽  
International Audience ◽  
Interval Trees ◽  
Number Of Leaves ◽  
Permutation Analysis

International audience After extending classical results on simple varieties of trees to trees counted by their number of leaves, we describe a filtration of the set of permutations based on their strong interval trees. For each subclass we provide asymptotic formulas for number of trees (by leaves), average number of nodes of fixed arity, average subtree size sum, and average number of internal nodes. The filtration is motivated by genome comparison of related species. Nous commençons par étendre les résultats classiques sur les variétés simples d'arbres aux arbres comptés selon leur nombre de feuilles, puis nous décrivons une filtration de l'ensemble des permutations qui repose sur leurs arbres des intervalles communs. Pour toute sous-classe, nous donnons des formules asymptotiques pour le nombre d'arbres (comptés selon les feuilles), le nombre moyen de nœuds d'arité fixée, la moyenne de la somme des tailles des sous-arbres, et le nombre moyen de nœuds internes. Cette filtration est motivée par des problématiques de comparaison de génomes.

scholarly journals Bernoulli trials and permutation statistics

1992 ◽  
Vol 15 (2) ◽  
pp. 291-311 ◽  
Author(s):  
Don Rawlings
Keyword(s):  
General Class ◽  
Binary Tree ◽  
Random Variables ◽  
Probability Measures ◽  
Permutation Statistics ◽  
Bernoulli Trials ◽  
Coin Tossing ◽  
Set Of Permutations ◽  
Natural Way

Several coin-tossing games are surveyed which, in a natural way, give rise to “statistically” induced probability measures on the set of permutations of{1,2,…,n}and on sets of multipermutations. The distributions of a general class of random variables known as binary tree statistics are also given.

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