scholarly journals Enumeration of Stack-Sorting Preimages via a Decomposition Lemma

2021 ◽  
Vol vol. 22 no. 2, Permutation... (Combinatorics) ◽  
Author(s):  
Colin Defant
Keyword(s):  
Explicit Formula ◽  
Catalan Numbers ◽  
Stack Sorting ◽  
Set Of Permutations ◽  
Order Ideals ◽  
Wilf Equivalence ◽  
Young’S Lattice

We give three applications of a recently-proven "Decomposition Lemma," which allows one to count preimages of certain sets of permutations under West's stack-sorting map $s$. We first enumerate the permutation class $s^{-1}(\text{Av}(231,321))=\text{Av}(2341,3241,45231)$, finding a new example of an unbalanced Wilf equivalence. This result is equivalent to the enumeration of permutations sortable by ${\bf B}\circ s$, where ${\bf B}$ is the bubble sort map. We then prove that the sets $s^{-1}(\text{Av}(231,312))$, $s^{-1}(\text{Av}(132,231))=\text{Av}(2341,1342,\underline{32}41,\underline{31}42)$, and $s^{-1}(\text{Av}(132,312))=\text{Av}(1342,3142,3412,34\underline{21})$ are counted by the so-called "Boolean-Catalan numbers," settling a conjecture of the current author and another conjecture of Hossain. This completes the enumerations of all sets of the form $s^{-1}(\text{Av}(\tau^{(1)},\ldots,\tau^{(r)}))$ for $\{\tau^{(1)},\ldots,\tau^{(r)}\}\subseteq S_3$ with the exception of the set $\{321\}$. We also find an explicit formula for $|s^{-1}(\text{Av}_{n,k}(231,312,321))|$, where $\text{Av}_{n,k}(231,312,321)$ is the set of permutations in $\text{Av}_n(231,312,321)$ with $k$ descents. This allows us to prove a conjectured identity involving Catalan numbers and order ideals in Young's lattice. Comment: 20 pages, 4 figures. arXiv admin note: text overlap with arXiv:1903.09138

scholarly journals On the Parity of Certain Coefficients for a $q$-Analogue of the Catalan Numbers

10.37236/2313 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Kendra Killpatrick
Keyword(s):  
Equivalence Classes ◽  
Catalan Numbers ◽  
Major Index ◽  
Powers Of 2 ◽  
Set Of Permutations ◽  
Adic Valuation ◽  
Permutation Statistic ◽  
Wilf Equivalence

The 2-adic valuation (highest power of 2) dividing the well-known Catalan numbers, $C_n$, has been completely determined by Alter and Kubota and further studied combinatorially by Deutsch and Sagan.  In particular, it is well known that $C_n$ is odd if and only if $n = 2^k-1$ for some $k \geq 0$.  The polynomial $F_n^{ch}(321;q) = \sum_{\sigma \in Av_n(321)} q^{ch(\sigma)}$, where $Av_n(321)$ is the set of permutations in $S_n$ that avoid 321 and $ch$ is the charge statistic, is a $q$-analogue of the Catalan numbers since specializing $q=1$ gives $C_n$.  We prove that the coefficient of $q^i$ in $F_{2^k-1}^{ch}(321;q)$ is even if $i \geq 1$, giving a refinement of the "if" direction of the $C_n$ parity result.  Furthermore, we use a bijection between the charge statistic and the major index to prove a conjecture of Dokos, Dwyer, Johnson, Sagan and Selsor regarding powers of 2 and the major index.    In addition, Sagan and Savage have recently defined a notion of $st$-Wilf equivalence for any permutation statistic $st$ and any two sets of permutations $\Pi$ and $\Pi'$.  We say $\Pi$ and $\Pi'$ are $st$-Wilf equivalent if $\sum_{\sigma \in Av_n(\Pi)} q^{st(\sigma)} = \sum_{\sigma \in Av_n(\Pi')} q^{st(\sigma)}$.  In this paper we show how one can characterize the charge-Wilf equivalence classes for subsets of $S_3$.

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 Matchings Avoiding Partial Patterns

10.37236/1138 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
William Y. C. Chen ◽  
Toufik Mansour ◽  
Sherry H. F. Yan
Keyword(s):  
Catalan Numbers ◽  
Schröder Numbers ◽  
Schröder Paths ◽  
Generating Trees ◽  
Super Catalan Numbers ◽  
Wilf Equivalence

We show that matchings avoiding a certain partial pattern are counted by the $3$-Catalan numbers. We give a characterization of $12312$-avoiding matchings in terms of restrictions on the corresponding oscillating tableaux. We also find a bijection between matchings avoiding both patterns $12312$ and $121323$ and Schröder paths without peaks at level one, which are counted by the super-Catalan numbers or the little Schröder numbers. A refinement of the super-Catalan numbers is derived by fixing the number of crossings in the matchings. In the sense of Wilf-equivalence, we use the method of generating trees to show that the patterns 12132, 12123, 12321, 12231, 12213 are all equivalent to the pattern $12312$.

scholarly journals Mixed Volumes of Hypersimplices

10.37236/5770 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Gaku Liu
Keyword(s):  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Type B ◽  
Combinatorial Interpretation ◽  
Mixed Volumes ◽  
Eulerian Numbers ◽  
Eulerian Number ◽  
Set Of Permutations

In this paper 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.

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 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 Wilf Classes of Pairs of Permutations of Length 4

10.37236/1922 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Ian Le
Keyword(s):  
Generating Function ◽  
Equivalence Classes ◽  
Order Relations ◽  
Set Of Permutations ◽  
Wilf Equivalence

$S_n(\pi_1,\pi_2,\dots, \pi_r)$ denotes the set of permutations of length $n$ that have no subsequence with the same order relations as any of the $\pi_i$. In this paper we show that $|S_n(1342,2143)|=|S_n(3142,2341)|$ and $|S_n(1342,3124)|=|S_n(1243,2134)|$. These two facts complete the classification of Wilf-equivalence classes for pairs of permutations of length four. In both instances we exhibit bijections between the sets using the idea of a "block", and in the former we find a generating function for $|S_n(1342,2143)|$.

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