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 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 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.

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 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 Emulation of Utility Functions Over a Set of Permutations: Sequencing Reliability Growth Tasks

Technometrics ◽  
2018 ◽  
Vol 60 (3) ◽  
pp. 273-285 ◽  
Author(s):  
Kevin J. Wilson ◽  
Daniel A. Henderson ◽  
John Quigley
Keyword(s):  
Utility Functions ◽  
Reliability Growth ◽  
Set Of Permutations

scholarly journals Quantifying Noninvertibility in Discrete Dynamical Systems

10.37236/9475 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Colin Defant ◽  
James Propp
Keyword(s):  
Dynamical Systems ◽  
Real Number ◽  
Arbitrary Function ◽  
Discrete Dynamical Systems ◽  
Open Problems ◽  
Natural Measure ◽  
Stack Sorting ◽  
Finite Set

Given a finite set $X$ and a function $f:X\to X$, we define the \emph{degree of noninvertibility} of $f$ to be $\displaystyle\deg(f)=\frac{1}{|X|}\sum_{x\in X}|f^{-1}(f(x))|$. This is a natural measure of how far the function $f$ is from being bijective. We compute the degrees of noninvertibility of some specific discrete dynamical systems, including the Carolina solitaire map, iterates of the bubble sort map acting on permutations, bubble sort acting on multiset permutations, and a map that we call "nibble sort." We also obtain estimates for the degrees of noninvertibility of West's stack-sorting map and the Bulgarian solitaire map. We then turn our attention to arbitrary functions and their iterates. In order to compare the degree of noninvertibility of an arbitrary function $f:X\to X$ with that of its iterate $f^k$, we prove that \[\max_{\substack{f:X\to X\\ |X|=n}}\frac{\deg(f^k)}{\deg(f)^\gamma}=\Theta(n^{1-1/2^{k-1}})\] for every real number $\gamma\geq 2-1/2^{k-1}$. We end with several conjectures and open problems.  

scholarly journals Equipopularity Classes in the Separable Permutations

10.37236/4797 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Michael Albert ◽  
Cheyne Homberger ◽  
Jay Pantone
Keyword(s):  
Set Of Permutations ◽  
Separable Permutations

When two patterns occur equally often in a set of permutations, we say that these patterns are equipopular. Using both structural and analytic tools, we classify the equipopular patterns in the set of separable permutations. In particular, we show that the number of equipopularity classes for length $n$ patterns in the separable permutations is equal to the number of partitions of $n-1$.

A Semilattice on the Set of Permutations on an Infinite Set

1974 ◽  
Vol 60 (1-6) ◽  
pp. 191-199
Author(s):  
I. G. Rosenberg
Keyword(s):  
Set Of Permutations ◽  
Infinite Set
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