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Distributions of Statistics over Pattern-Avoiding Permutations [article]

Michael Bukata, Ryan Kulwicki, Nicholas Lewandowski, Lara Pudwell, Jacob Roth, Teresa Wheeland
2019 arXiv   pre-print
Many of these statistics have already been studied over sets of permutations avoiding a single pattern of length 3.  ...  However, the distribution of peaks over 321-avoiding permutations is new and we relate it statistics on Dyck paths.  ...  of statistics over pattern-avoiding permutations whether we consider them over all Dyck paths or only over indecomposable Dyck paths  ... 
arXiv:1812.07112v3 fatcat:bptygbyjj5anhgjk5xfkvx6a3e

The equidistribution of some Mahonian statistics over permutations avoiding a pattern of length three [article]

Phan Thuan Do, Thi Thu Huong Tran, Vincent Vajnovszki
2021 arXiv   pre-print
We prove the equidistribution of several multistatistics over some classes of permutations avoiding a 3-length pattern.  ...  We deduce the equidistribution, on the one hand of inv and foze" statistics, and on the other hand that of maj and makl statistics, over these classes of pattern avoiding permutations.  ...  The authors are grateful to the anonymous referees for their careful reading of the paper and for providing many useful suggestions.  ... 
arXiv:2103.10357v2 fatcat:rgilfhsqd5as7a2il4rtphsxoe

Refining enumeration schemes to count according to permutation statistics [article]

Andrew M. Baxter
2014 arXiv   pre-print
We consider the question of computing the distribution of a permutation statistics over restricted permutations via enumeration schemes.  ...  We prove that when a finite enumeration scheme exists to compute the number of permutations avoiding a given set of vincular patterns, the scheme can also compute the distribution of certain permutation  ...  In this paper, we discuss how to refine enumeration schemes to compute the distributions of certain permutation statistics over a set of pattern-avoiding permutations.  ... 
arXiv:1401.0337v1 fatcat:pvkra3bemfdvblyeiuc4ys7qqu

Refined Wilf-equivalences by Comtet statistics

Shishuo Fu, Zhicong Lin, Yaling Wang
2021 Electronic Research Archive  
This work is motivated by a triple equidistribution result of Rubey on 321-avoiding permutations, and a recent result of the first and third authors that iar is a Comtet statistic over separable permutations  ...  Calculations of the pdes, iar, compq generating functions over these pattern avoiding classes and separable permutations. ' A further refinement of Wang's descent-double descent-Wilf equivalence between  ...  Part of this work was initiated at the Research Center for Mathematics and Interdisciplinary Sciences at Shandong University.  ... 
doi:10.3934/era.2021018 fatcat:lgtiq2a4zreq5aunx46t7rtv2i

Equidistributions of MAJ and STAT over pattern avoiding permutations [article]

Joanna N. Chen
2017 arXiv   pre-print
Recently, Amini investigated the equidistributions of these Mahonian statistics over sets of pattern avoiding permutations. Moreover, he posed several conjectures.  ...  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  ...  Recently, Amini [1] studied the distributions of these Mahonian statistics defined by combinations of generalized pattern functions at most three over sets of pattern avoiding permutations.  ... 
arXiv:1707.07195v1 fatcat:rc7akbswevay3llxs36vqfv5j4

The descent statistic over 123-avoiding permutations [article]

M.Barnabei, F.Bonetti, M.Silimbani
2009 arXiv   pre-print
We exploit Krattenthaler's bijection between 123-avoiding permutations and Dyck paths to determine the Eulerian distribution over the set S_n(123) of 123-avoiding permutations in S_n.  ...  We get the Eulerian numbers of S_n(123) by studying the joint distribution of these two statistics on Dyck paths.  ...  The subset of S n of all permutations avoiding a pattern τ is denoted by S n (τ ). Pattern avoiding permutations have been intensively studied in recent years from many points of view (see e.g.  ... 
arXiv:0910.0963v1 fatcat:3qkhav6vozcthgzxmw4ejq6u7y

Generalized permutation patterns -- a short survey [article]

Einar Steingrimsson
2008 arXiv   pre-print
Subsets of permutations characterized by the avoidance--or the prescribed number of occurrences--of generalized patterns exhibit connections to an enormous variety of other combinatorial structures, some  ...  In an occurrence of a generalized pattern, some letters of that subsequence may be required to be adjacent in the permutation.  ...  A permutation statistic is Mahonian if it has the same distribution-on the set of permutations of length n for each n-as the number of inversions.  ... 
arXiv:0801.2412v2 fatcat:hgo4jp2z75fftjj34u4hxylc5u

Major index distribution over permutation classes [article]

Michal Opler
2015 arXiv   pre-print
In this paper, we study the distribution of the major index over pattern-avoiding permutations of length n.  ...  We focus on the number M_n^m(Π) of permutations of length n with major index m and avoiding the set of patterns Π.  ...  Monotonicity of columns In this section, we will focus on the distribution of major index over permutations avoiding a single pattern.  ... 
arXiv:1505.07135v1 fatcat:dl6ykwhcdzarphmf3fmdyspwru

Statistics on Almost-Fibonacci Pattern-Avoiding Permutations [article]

Brody Lynch, Yihan Qin
2022 arXiv   pre-print
all permutations of length n that avoid all of the patterns in the set S.  ...  Finally, we encode these statistics in generating functions written in terms of the generating function for Fibonacci permutations.  ...  We can now consider the distribution of this statistic over permutations in each set. Theorem 3.7.  ... 
arXiv:2203.11416v1 fatcat:52fskumj5vgxxdureaiff56pme

Combinatorial statistics on type-B analogues of noncrossing partitions and restricted permutations

Rodica Simion
2000 Electronic Journal of Combinatorics  
We also identify pattern-avoiding classes of elements in the hyperoctahedral group which parallel known classes of restricted permutations with respect to their relations to noncrossing partitions.  ...  We define type-B analogues of combinatorial statistics previously studied on noncrossing partitions and show that analogous equidistribution and symmetry properties hold in the case of type-B noncrossing  ...  The descent statistic and the major index statistic of σ are des A (σ): = #Des A (σ), maj A (σ): = i∈Des A (σ) i. (19) Pairs of permutation statistics whose joint distribution over S n coincides with that  ... 
doi:10.37236/1487 fatcat:5qr4hxa2hfhjhow2sa3pycm2fq

Generalized Pattern Avoidance

Anders Claesson
2001 European journal of combinatorics (Print)  
We will consider pattern avoidance for such patterns, and give a complete solution for the number of permutations avoiding any single pattern of length three with exactly one adjacent pair of letters.  ...  We also give some results for the number of permutations avoiding two different patterns.  ...  It is well known (see [2] ) that the distribution of R over all Dyck paths of length 2n is the distribution we claim that L has over S n (b ac). .  ... 
doi:10.1006/eujc.2001.0515 fatcat:q43shpidzfftdaqlkgd2nldqeu

Set Partition Patterns and the Dimension Index [article]

Thomas Grubb, Frederick Rajasekaran
2020 arXiv   pre-print
We continue this work by computing the distribution of the dimension index (a statistic arising from the supercharacter theory of finite groups) across certain avoidance classes of partitions.  ...  Work of Sagan and of Goyt has led to enumerative results in avoidance classes of set partitions, which were refined by Dahlberg et al. through the use of combinatorial statistics.  ...  Longer Patterns: The most obvious extension would could make to this article is to continue studying the distribution of these statistics over avoidance classes of longer patterns.  ... 
arXiv:2009.00650v1 fatcat:cypju4czljaolp4kxeh66f2vfq

On 021-Avoiding Ascent Sequences [article]

William Y. C. Chen, Alvin Y. L. Dai, Theodore Dokos, Tim Dwyer, Bruce E. Sagan
2012 arXiv   pre-print
Duncan and Steingrímsson have shown that the ascent statistic has the same distribution over _n(021) as over S_n(132).  ...  We let _n stand for the set of such sequences and use _n(p) for the subset of sequences avoiding a pattern p.  ...  This work was supported by the 973 Project, the PCSIRT Project of the Ministry of Education, and the National Science Foundation of China.  ... 
arXiv:1206.2849v2 fatcat:6gku7f4isvcrflyqk3a5ogzaea

Segmental partially ordered generalized patterns

Sergey Kitaev
2005 Theoretical Computer Science  
The entire distribution of the maximum number of non-overlapping occurrences of a generalized pattern with no dashes in permutations or words studied in [S.  ...  Steingrímsson, Generalized permutation patterns and a classification of the Mahonian statistics, Séminaire Lotharingien de Combinatoire, 2000, B44b:18pp] for permutations and in [A. Burstein, T.  ...  He is also grateful to the referee for useful remarks and for providing a simpler proof of Proposition 9.  ... 
doi:10.1016/j.tcs.2005.09.036 fatcat:6eu3xguxe5f73evcv5h6xl4f2m

Patterns in Random Permutations Avoiding Some Other Patterns (Keynote Speakers)

Svante Janson, Marc Herbstritt
2018 International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms  
Consider a random permutation drawn from the set of permutations of length n that avoid a given set of one or several patterns of length 3.  ...  We show that the number of occurrences of another pattern has a limit distribution, after suitable scaling.  ...  A o f A 2 0 1 8 6:4 Patterns in Random Permutations Avoiding Some Other Patterns Remark.  ... 
doi:10.4230/lipics.aofa.2018.6 dblp:conf/aofa/Janson18 fatcat:qensxeqtcngp7mkf5okgx25jsi
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