scholarly journals A New Bijective Proof of Babson and Steingrímsson's Conjecture

10.37236/6411 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Joanna N. Chen ◽  
Shouxiao Li
Keyword(s):  
Generating Functions ◽  
Permutation Patterns ◽  
Bijective Proof ◽  
Major Index ◽  
Mahonian Statistics ◽  
Pattern Functions

Babson and Steingrímsson introduced generalized permutation patterns and showed that most of the Mahonian statistics in the literature can be expressed by the combination of generalized pattern functions. Particularly, they defined a new Mahonian statistic in terms of generalized pattern functions, which is denoted $stat$. Given a permutation $\pi$, let $des(\pi)$ denote the descent number of $\pi$ and $maj(\pi)$ denote the major index of $\pi$. Babson and Steingrímsson conjectured that $(des,stat)$ and $(des,maj)$ are equidistributed on $S_n$. Foata and Zeilberger settled this conjecture using q-enumeration, generating functions and Maple packages ROTA and PERCY. Later, Burstein provided a bijective proof of a refinement of this conjecture. In this paper, we give a new bijective proof of this conjecture.

scholarly journals Inversion Generating Functions for Signed Pattern Avoiding Permutations

10.37236/6545 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Naiomi T. Cameron ◽  
Kendra Killpatrick
Keyword(s):  
Generating Functions ◽  
Combinatorial Proof ◽  
Hyperoctahedral Group ◽  
Major Index ◽  
Signed Permutations ◽  
Mahonian Statistics ◽  
Almost All ◽  
Open Question ◽  
Pattern Avoiding Permutations

We consider the classical Mahonian statistics on the set $B_n(\Sigma)$ of signed permutations in the hyperoctahedral group $B_n$ which avoid all patterns in $\Sigma$, where $\Sigma$ is a set of patterns of length two.  In 2000, Simion gave the cardinality of $B_n(\Sigma)$ in the cases where $\Sigma$ contains either one or two patterns of length two and showed that $\left|B_n(\Sigma)\right|$ is constant whenever $\left|\Sigma\right|=1$, whereas in most but not all instances where $\left|\Sigma\right|=2$, $\left|B_n(\Sigma)\right|=(n+1)!$.  We answer an open question of Simion by providing bijections from $B_n(\Sigma)$ to $S_{n+1}$ in these cases where $\left|B_n(\Sigma)\right|=(n+1)!$.  In addition, we extend Simion's work by providing a combinatorial proof in the language of signed permutations for the major index on $B_n(21, \bar{2}\bar{1})$ and by giving the major index on $D_n(\Sigma)$ for $\Sigma =\{21, \bar{2}\bar{1}\}$ and $\Sigma=\{12,21\}$.  The main result of this paper is to give the inversion generating functions for $B_n(\Sigma)$ for almost all sets $\Sigma$ with $\left|\Sigma\right|\leq2.$

scholarly journals Odd and Even Major Indices and One-Dimensional Characters for Classical Weyl Groups

2020 ◽  
Vol 24 (4) ◽  
pp. 809-835
Author(s):  
Francesco Brenti ◽  
Paolo Sentinelli
Keyword(s):  
Generating Functions ◽  
Weyl Groups ◽  
One Dimensional ◽  
Major Index ◽  
The One

Abstract We define and study odd and even analogues of the major index statistics for the classical Weyl groups. More precisely, we show that the generating functions of these statistics, twisted by the one-dimensional characters of the corresponding groups, always factor in an explicit way. In particular, we obtain odd and even analogues of Carlitz’s identity, of the Gessel–Simion Theorem, and a parabolic extension, and refinement, of a result of Wachs.

scholarly journals Generalized Pattern Avoidance Condition for the Wreath Product of Cyclic Groups with Symmetric Groups

2013 ◽  
Vol 2013 ◽  
pp. 1-17
Author(s):  
Sergey Kitaev ◽  
Jeffrey Remmel ◽  
Manda Riehl
Keyword(s):  
Symmetric Group ◽  
Wreath Product ◽  
Cyclic Group ◽  
Generating Functions ◽  
Symmetric Groups ◽  
Pattern Avoidance ◽  
Catalan Numbers ◽  
Cyclic Groups ◽  
Bijective Proof ◽  
Avoidance Condition

We continue the study of the generalized pattern avoidance condition for Ck≀Sn, the wreath product of the cyclic group Ck with the symmetric group Sn, initiated in the work by Kitaev et al., In press. Among our results, there are a number of (multivariable) generating functions both for consecutive and nonconsecutive patterns, as well as a bijective proof for a new sequence counted by the Catalan numbers.

scholarly journals 0-Hecke algebra actions on coinvariants and flags

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Jia Huang
Keyword(s):  
Generating Functions ◽  
Hecke Algebra ◽  
Flag Variety ◽  
Major Index ◽  
Inversion Number ◽  
International Audience ◽  
Descent Set ◽  
Complete Flag ◽  
Coinvariant Algebra

International audience By investigating the action of the 0-Hecke algebra on the coinvariant algebra and the complete flag variety, we interpret generating functions counting the permutations with fixed inverse descent set by their inversion number and major index. En étudiant l'action de l'algèbre de 0-Hecke sur l'algèbre coinvariante et la variété de drapeaux complète, nous interprétons les fonctions génératrices qui comptent les permutations avec un ensemble inverse de descentes fixé, selon leur nombre d'inversions et leur "major index''.

scholarly journals Mesh Patterns and the Expansion of Permutation Statistics as Sums of Permutation Patterns

10.37236/2001 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Petter Brändén ◽  
Anders Claesson
Keyword(s):  
Fixed Points ◽  
Relative Position ◽  
Permutation Statistics ◽  
Permutation Patterns ◽  
Alternating Permutations ◽  
Major Index ◽  
Permutation Pattern ◽  
New Type ◽  
Permutation Statistic ◽  
Infinite Linear

Any permutation statistic $f:{\mathfrak{S}}\to{\mathbb C}$ may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: $f= \Sigma_\tau\lambda_f(\tau)\tau$. To provide explicit expansions for certain statistics, we introduce a new type of permutation patterns that we call mesh patterns. Intuitively, an occurrence of the mesh pattern $p=(\pi,R)$ is an occurrence of the permutation pattern $\pi$ with additional restrictions specified by $R$ on the relative position of the entries of the occurrence. We show that, for any mesh pattern $p=(\pi,R)$, we have $\lambda_p(\tau) = (-1)^{|\tau|-|\pi|}{p}^{\star}(\tau)$ where ${p}^{\star}=(\pi,R^c)$ is the mesh pattern with the same underlying permutation as $p$ but with complementary restrictions. We use this result to expand some well known permutation statistics, such as the number of left-to-right maxima, descents, excedances, fixed points, strong fixed points, and the major index. We also show that alternating permutations, André permutations of the first kind and simsun permutations occur naturally as permutations avoiding certain mesh patterns. Finally, we provide new natural Mahonian statistics.

scholarly journals Pattern Avoidance in Matchings and Partitions

10.37236/2976 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Jonathan Bloom ◽  
Sergi Elizalde
Keyword(s):  
Fixed Points ◽  
Generating Functions ◽  
Direct Proof ◽  
Pattern Avoidance ◽  
Equivalence Classes ◽  
Set Partitions ◽  
Bijective Proof ◽  
Rook Placements ◽  
Wilf Equivalence ◽  
Diagram Representation

Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize $3$-crossings and $3$-nestings, have an interpretation, in the case of matchings, in terms of patterns in full rook placements on Ferrers boards.We enumerate $312$-avoiding matchings and partitions, obtaining algebraic generating functions, in contrast with the known D-finite generating functions for the $321$-avoiding (i.e., $3$-noncrossing) case. Our approach provides a more direct proof of a formula of Bóna for the number of $1342$-avoiding permutations. We also give a bijective proof of the shape-Wilf-equivalence of the patterns $321$ and $213$ which greatly simplifies existing proofs by Backelin-West-Xin and Jelínek, and provides an extension of work of Gouyou-Beauchamps for matchings with fixed points. Finally, we classify pairs of patterns of length 3 according to shape-Wilf-equivalence, and enumerate matchings and partitions avoiding a pair in most of the resulting equivalence classes.

scholarly journals Signed Words and Permutations II; The Euler-Mahonian Polynomials

10.37236/1879 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
Dominique Foata ◽  
Guo-Niu Han
Keyword(s):  
Symmetric Group ◽  
Generating Functions ◽  
Statistical Study ◽  
Record Values ◽  
The Other ◽  
Length Function ◽  
Major Index ◽  
Signed Permutations ◽  
Inversion Number ◽  
The One

As for the symmetric group of ordinary permutations there is also a statistical study of the group of signed permutations, that consists of calculating multivariable generating functions for this group by statistics involving record values and the length function. Two approaches are here systematically explored, using the flag-major index on the one hand, and the flag-inversion number on the other hand. The MacMahon Verfahren appears as a powerful tool throughout.

scholarly journals Graphical Mahonian Statistics on Words

10.37236/6263 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Amy Grady ◽  
Svetlana Poznanović
Keyword(s):  
Directed Graphs ◽  
Permutation Statistics ◽  
Major Index ◽  
Mahonian Statistics

Foata and Zeilberger defined the graphical major index, $\mathrm{maj}_U$, and the graphical inversion index, $\mathrm{inv}_U$, for words over the alphabet $\{1, 2, \dots, n\}$. These statistics are a generalization of the classical permutation statistics $\mathrm{maj}$ and $\mathrm{inv}$ indexed by directed graphs $U$. They showed that $\mathrm{maj}_U$ and $\mathrm{inv}_U$ are equidistributed over all rearrangement classes if and only if $U$ is bipartitional. In this paper we strengthen their result by showing that if $\mathrm{maj}_U$ and $\mathrm{inv}_U$ are equidistributed on a single rearrangement class then $U$ is essentially bipartitional. Moreover, we define a graphical sorting index, $\mathrm{sor}_U$, which generalizes the sorting index of a permutation. We then characterize the graphs $U$ for which $\mathrm{sor}_U$ is equidistributed with $\mathrm{inv}_U$ and $\mathrm{maj}_U$ on a single rearrangement class. 

scholarly journals Hopf algebra of permutation pattern functions

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Yannic Vargas
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Point Of View ◽  
Algebraic Combinatorics ◽  
Permutation Patterns ◽  
Permutation Pattern ◽  
International Audience ◽  
Pattern Functions

International audience We study permutation patterns from an algebraic combinatorics point of view. Using analogues of the classical shuffle and infiltration products for word, we define two new Hopf algebras of permutations related to the notion of permutation pattern. We show several remarkable properties of permutation patterns functions, as well their occurrence in other domains.

scholarly journals Rhombic alternative tableaux, assemblees of permutations, and the ASEP

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Olya Mandelshtam ◽  
Xavier Viennot
Keyword(s):  
Steady State ◽  
Generating Function ◽  
Generating Functions ◽  
Finite Lattice ◽  
One Dimensional ◽  
Open Boundaries ◽  
Bijective Proof ◽  
Insertion Algorithm ◽  
International Audience ◽  
Steady State Probabilities

International audience In this paper, we introduce therhombic alternative tableaux, whose weight generating functions providecombinatorial formulae to compute the steady state probabilities of the two-species ASEP. In the ASEP, there aretwo species of particles, oneheavyand onelight, on a one-dimensional finite lattice with open boundaries, and theparametersα,β, andqdescribe the hopping probabilities. The rhombic alternative tableaux are enumerated by theLah numbers, which also enumerate certainassembl ́ees of permutations. We describe a bijection between the rhombicalternative tableaux and these assembl ́ees. We also provide an insertion algorithm that gives a weight generatingfunction for the assemb ́ees. Combined, these results give a bijective proof for the weight generating function for therhombic alternative tableaux.

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