scholarly journals On a Refinement of Wilf-equivalence for Permutations

10.37236/4465 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Sherry H.F. Yan ◽  
Huiyun Ge ◽  
Yaqiu Zhang
Keyword(s):  
Descent Set ◽  
Wilf Equivalence

Recently, Dokos et al. conjectured that for all $k, m\geq 1$, the patterns $ 12\ldots k(k+m+1)\ldots (k+2)(k+1) $ and $(m+1)(m+2)\ldots  (k+m+1)m\ldots 21$ are $maj$-Wilf-equivalent. In this paper, we confirm this conjecture for all $k\geq 1$ and $m=1$. In fact, we construct a descent set preserving bijection between $ 12\ldots k (k-1) $-avoiding permutations and $23\ldots k1$-avoiding permutations for all $k\geq 3$. As a corollary, our bijection enables us to settle a conjecture of  Gowravaram and Jagadeesan concerning the Wilf-equivalence for  permutations with given descent sets.

scholarly journals A General Theory of Wilf-Equivalence for Catalan Structures

10.37236/5629 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Michael Albert ◽  
Mathilde Bouvel
Keyword(s):  
General Theory ◽  
Equivalence Relation ◽  
Generating Functions ◽  
Asymptotic Estimate ◽  
Equivalence Classes ◽  
Catalan Number ◽  
Combinatorial Structures ◽  
Dyck Paths ◽  
Significant Interest ◽  
Wilf Equivalence

The existence of apparently coincidental equalities (also called Wilf-equivalences) between the enumeration sequences or generating functions of various hereditary classes of combinatorial structures has attracted significant interest. We investigate such coincidences among non-crossing matchings and a variety of other Catalan structures including Dyck paths, 231-avoiding permutations and plane forests. In particular we consider principal subclasses defined by not containing an occurrence of a single given structure. An easily computed equivalence relation among structures is described such that if two structures are equivalent then the associated principal subclasses have the same enumeration sequence. We give an asymptotic estimate of the number of equivalence classes of this relation among structures of size $n$ and show that it is exponentially smaller than the $n^{th}$ Catalan number. In other words these "coincidental" equalities are in fact very common among principal subclasses. Our results also allow us to prove in a unified and bijective manner several known Wilf-equivalences from the literature.

scholarly journals Wilf equivalence relations for consecutive patterns

2018 ◽  
Vol 99 ◽  
pp. 134-157 ◽  
Author(s):  
Tim Dwyer ◽  
Sergi Elizalde
Keyword(s):  
Equivalence Relations ◽  
Wilf Equivalence

scholarly journals Wilf-equivalence for singleton classes

2007 ◽  
Vol 38 (2) ◽  
pp. 133-148 ◽  
Author(s):  
Jörgen Backelin ◽  
Julian West ◽  
Guoce Xin
Keyword(s):  
Wilf Equivalence

scholarly journals The signed descent set polynomial revisited

2016 ◽  
Vol 339 (9) ◽  
pp. 2263-2266 ◽  
Author(s):  
Richard Ehrenborg ◽  
N. Bradley Fox
Keyword(s):  
Descent Set

scholarly journals Rook and Wilf equivalence of integer partitions

2018 ◽  
Vol 71 ◽  
pp. 246-267 ◽  
Author(s):  
Jonathan Bloom ◽  
Dan Saracino
Keyword(s):  
Integer Partitions ◽  
Wilf Equivalence

scholarly journals Descents and des-Wilf equivalence of permutations avoiding certain nonclassical patterns

2019 ◽  
Vol 12 (4) ◽  
pp. 549-563
Author(s):  
Caden Bielawa ◽  
Robert Davis ◽  
Daniel Greeson ◽  
Qinhan Zhou
Keyword(s):  
Wilf Equivalence

scholarly journals Variations on Descents and Inversions in Permutations

10.37236/856 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Denis Chebikin
Keyword(s):  
Generating Functions ◽  
Euler Number ◽  
Weighted Sum ◽  
Eulerian Polynomials ◽  
Joint Distributions ◽  
Dyck Paths ◽  
Sigma 1 ◽  
Descent Set ◽  
New Statistics ◽  
Factor Set

We study new statistics on permutations that are variations on the descent and the inversion statistics. In particular, we consider the alternating descent set of a permutation $\sigma = \sigma_1\sigma_2\cdots\sigma_n$ defined as the set of indices $i$ such that either $i$ is odd and $\sigma_i > \sigma_{i+1}$, or $i$ is even and $\sigma_i < \sigma_{i+1}$. We show that this statistic is equidistributed with the odd $3$-factor set statistic on permutations $\tilde{\sigma} = \sigma_1\sigma_2\cdots\sigma_{n+1}$ with $\sigma_1=1$, defined to be the set of indices $i$ such that the triple $\sigma_i \sigma_{i+1} \sigma_{i+2}$ forms an odd permutation of size $3$. We then introduce Mahonian inversion statistics corresponding to the two new variations of descents and show that the joint distributions of the resulting descent-inversion pairs are the same, establishing a connection to two classical Mahonian statistics, maj and stat, along the way. We examine the generating functions involving alternating Eulerian polynomials, defined by analogy with the classical Eulerian polynomials $\sum_{\sigma\in\mathcal{S}_n} t^{{\rm des}(\sigma)+1}$ using alternating descents. For the alternating descent set statistic, we define the generating polynomial in two non-commutative variables by analogy with the $ab$-index of the Boolean algebra $B_n$, providing a link to permutations without consecutive descents. By looking at the number of alternating inversions, which we define in the paper, in alternating (down-up) permutations, we obtain a new $q$-analog of the Euler number $E_n$ and show how it emerges in a $q$-analog of an identity expressing $E_n$ as a weighted sum of Dyck paths.

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