Preimages under the Queuesort algorithm

Preimages under the bubblesort operator

We study preimages of permutations under the bubblesort operator $\mathbf{B}$. We achieve a description of these preimages much more complete than what is known for the more complicated sorting operators $\mathbf{S}$ (stacksort) and $\mathbf{Q}$ (queuesort). We describe explicitly the set of preimages under $\mathbf{B}$ of any permutation $π$ from the left-to-right maxima of $π$, showing that there are $2^{k-1}$ such preimages if $k$ is the number of these left-to-right maxima. We further consider, for each $n$, the tree $T_n$ recording all permutations of size $n$ in its nodes, in which an ed…

Preimages under the Stack-Sorting Algorithm

We use a method for determining the number of preimages of any permutation under the stack-sorting map in order to obtain recursive upper bounds for the numbers $W_t(n)$ and $W_t(n,k)$ of $t$-stack sortable permutations of length $n$ and $t$-stack sortable permutations of length $n$ with exactly $k$ descents. From these bounds, we are able to significantly improve the best known upper bounds for $\displaystyle{\lim_{n\to\infty}\sqrt[n]{W_t(n)}}$ when $t=3$ and $t=4$.

Coalescence under Preimage Constraints

The primary goal of this document is to record the asymptotic effects that preimage constraints impose upon the sizes of the iterated images of a random function. Specifically, given a subset $\mathcal{P}\subseteq \mathbb{Z}_{\geq 0}$ and a finite set $S$ of size $n$, choose a function uniformly from the set of functions $f:S\rightarrow S$ that satisfy the condition that $|f^{-1}(x)|\in\mathcal{P}$ for each $x\in S$, and ask what $|f^k(S)|$ looks like as $n$ goes to infinity. The robust theory of singularity analysis allows one to completely answer this question if one accepts that $0\in\mathc…

Restricted Stacks as Functions

The stack sort algorithm has been the subject of extensive study over the years. In this paper we explore a generalized version of this algorithm where instead of avoiding a single decrease, the stack avoids a set $T$ of permutations. We let $s_T$ denote this map. We classify for which sets $T$ the map $s_T$ is bijective. A corollary to this answers a question of Baril, Cerbai, Khalil, and Vajnovszki about stack sort composed with $s_{\{σ,τ\}}$, known as the $(σ,τ)$-machine. This fully classifies for which $σ$ and $τ$ the preimage of the identity under the $(σ,τ)$-machine is counted by the Cat…

通过分解引理枚举栈排序前像

Enumeration of Stack-Sorting Preimages via a Decomposition Lemma

本文利用分解引理枚举了West栈排序映射下若干排列类的前像，包括发现不平衡Wilf等价、验证布尔-加泰兰数计数猜想，并给出带下降数限制的显式公式，从而证明涉及加泰兰数与Young格序理想的恒等式。

Fertility Numbers

A nonnegative integer is called a fertility number if it is equal to the number of preimages of a permutation under West's stack-sorting map. We prove structural results concerning permutations, allowing us to deduce information about the set of fertility numbers. In particular, the set of fertility numbers is closed under multiplication and contains every nonnegative integer that is not congruent to $3$ modulo $4$. We show that the lower asymptotic density of the set of fertility numbers is at least $1954/2565\approx 0.7618$. We also exhibit some positive integers that are not fertility numbe…