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Register a new userStack-Sorting with Consecutive-Pattern-Avoiding Stacks
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We introduce consecutive-pattern-avoiding stack-sorting maps $text{SC}_sigma$, which are natural generalizations of Wests stack-sorting map $s$ and natural analogues of the classical-pattern-avoiding stack-sorting maps $s_sigma$ recently introduced by Cerbai, Claesson, and Ferrari. We characterize the patterns $sigma$ such that $text{Sort}(text{SC}_sigma)$, the set of permutations that are sortable via the map $scirctext{SC}_sigma$, is a permutation class, and we enumerate the sets $text{Sort}(text{SC}_{sigma})$ for $sigmain{123,132,321}$. We also study the maps $text{SC}_sigma$ from a dynamical point of view, characterizing the periodic points of $text{SC}_sigma$ for all $sigmain S_3$ and computing $max_{piin S_n}|text{SC}_sigma^{-1}(pi)|$ for all $sigmain{132,213,231,312}$. In addition, we characterize the periodic points of the classical-pattern-avoiding stack-sorting map $s_{132}$, and we show that the maximum number of iterations of $s_{132}$ needed to send a permutation in $S_n$ to a periodic point is $n-1$. The paper ends with numerous open problems and conjectures.
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A emph{set partition} of the set $[n]={1,...c,n}$ is a collection of disjoint blocks $B_1,B_2,...c, B_d$ whose union is $[n]$. We choose the ordering of the blocks so that they satisfy $min B_1<min B_2<...b<min B_d$. We represent such a set partition by a emph{canonical sequence} $pi_1,pi_2,...c,pi_n$, with $pi_i=j$ if $iin B_j$. We say that a partition $pi$ emph{contains} a partition $sigma$ if the canonical sequence of $pi$ contains a subsequence that is order-isomorphic to the canonical sequence of $sigma$. Two partitions $sigma$ and $sigma$ are emph{equivalent}, if there is a size-preserving bijection between $sigma$-avoiding and $sigma$-avoiding partitions. We determine several infinite families of sets of equivalent patterns; for instance, we prove that there is a bijection between $k$-noncrossing and $k$-nonnesting partitions, with a notion of crossing and nesting based on the canonical sequence. We also provide new combinatorial interpretations of the Catalan numbers and the Stirling numbers. Using a systematic computer search, we verify that our results characterize all the pairs of equivalent partitions of size at most seven. We also present a correspondence between set partitions and fillings of Ferrers shapes and stack polyominoes. This correspondence allows us to apply recent results on polyomino fillings in the study of partitions, and conversely, some of our results on partitions imply new results on polyomino fillings and ordered graphs.
An alternating permutation of length $n$ is a permutation $pi=pi_1 pi_2 ... pi_n$ such that $pi_1 < pi_2 > pi_3 < pi_4 > ...$. Let $A_n$ denote set of alternating permutations of ${1,2,..., n}$, and let $A_n(sigma)$ be set of alternating permutations in $A_n$ that avoid a pattern $sigma$. Recently, Lewis used generating trees to enumerate $A_{2n}(1234)$, $A_{2n}(2143)$ and $A_{2n+1}(2143)$, and he posed several conjectures on the Wilf-equivalence of alternating permutations avoiding certain patterns. Some of these conjectures have been proved by Bona, Xu and Yan. In this paper, we prove the two relations $|A_{2n+1}(1243)|=|A_{2n+1}(2143)|$ and $|A_{2n}(4312)|=|A_{2n}(1234)|$ as conjectured by Lewis.
Babson and Steingr{i}msson 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 $stat$. Recently, Amini investigated the equidistributions of these Mahonian statistics over sets of pattern avoiding permutations. Moreover, he posed several conjectures. In this paper, we construct a bijection from $S_n(213)$ to $S_n(231)$, which maps the statistic $(maj,stat)$ to the statistic $(stat,maj)$. This allows us to give solutions to some of Aminis conjectures.
In this note, we study the mean length of the longest increasing subsequence of a uniformly sampled involution that avoids the pattern $3412$ and another pattern.
The notion of a $p$-Riordan graph generalizes that of a Riordan graph, which, in turn, generalizes the notions of a Pascal graph and a Toeplitz graph. In this paper we introduce the notion of a $p$-Riordan word, and show how to encode $p$-Riordan graphs by $p$-Riordan words. For special important cases of Riordan graphs (the case $p=2$) and oriented Riordan graphs (the case $p=3$) we provide alternative encodings in terms of pattern-avoiding permutations and certain balanced words, respectively. As a bi-product of our studies, we provide an alternative proof of a known enumerative result on closed walks in the cube.
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