62 problems

Let $\mathfrak{S}_n$ be the set of permutations of $[n]$, let $\mathcal{S}$ be the stack-sorting operation, and let $\mathfrak{S}_n^2$ denote the permutations sorted by applying…

Let $\mathcal{W}'_2(n)$ denote the set of objects defined in the paper for which the enumeration $|\mathcal{W}'_2(n)|$ is considered, and let the $n$th Motzkin number be the Motzki…

Let $\mathcal D_n$ be the average number of iterations of the ordinary stack-sorting map $\operatorname{\mathtt{s}}$ needed to send an element of $S_n$ to the identity permutation,…

Let $\operatorname{rev}$ reverse a permutation, and define $\operatorname{revstack}=s\circ\operatorname{rev}$, where $s$ is the stack-sorting map. For $n\geq 1$ and $0\leq t<n$, le…

Let $W_3(n)$ denote the number of 3-stack-sortable permutations of length $n$. Growth-rate conjecture. The exponential growth rate satisfies … The bounds were proposed from numeric…

Sublinear minimal permutation size conjecture. We have

Monotonicity conjecture. The sequence

West's convergence conjecture. The quotient $\mathcal{D}_n/n$ converges as $n$ approaches infinity.

Polynomial correction conjecture. For each positive integer $k \geq 3$, there exists a finite positive integer $c$ and a degree $k-2$ polynomial $p(m)$ such that for all $m \geq c$…

Let $m(n)$ be the maximum sort-number among permutations of length $n$, where the sort-number is the number of iterations of $SC_{231}$ needed to reach the relevant sorted class. M…

Let $a(n)$ be the average sort-number of permutations of length $n$. Power-law exponent conjecture. Find the value of $k$ such that … is finite, or prove that no such value exists.…

Let $a(n)$ be the average sort-number of permutations of length $n$, where the sort-number is the number of iterations of $SC_{231}$ needed to reach the relevant sorted class. Aver…

Let $S_n$ be the symmetric group on $n$ letters, and let $\operatorname{Av}_n(132,231)$ denote the set of permutations in $S_n$ avoiding the patterns $132$ and $231$. For a permuta…

Let $\boldsymbol{\pi}\in\mathfrak{S}_n$ be a permutation, and consider the stack-sorting polytope generated by $\boldsymbol{\pi}$. The source identifies the volume of the polytope…

Let $t\geq 1$, let $p$ be either $213$ or $321$, and call a permutation $p$-avoiding if it contains no subsequence order-isomorphic to $p$. Let $t$-stack-sortable permutations be t…

Let $\mathfrak{S}_n^t$ be the set of permutations of length $n$ that become the identity after $t$ applications of the stack-sorting operation, and let $\mathfrak{S}_n^t(p)$ denote…

Let $t\geq 1$, $n\geq 0$, and $p\in\{213,321\}$. A permutation is $t$-stack-sortable if it can be sorted using $t$ applications of the stack-sorting operator, and it avoids a patte…

Let $t\geq 0$ and $n\geq 0$. A permutation is $t$-stack-sortable if it can be sorted using $t$ applications of the stack-sorting operator, and a permutation avoids a pattern if it…

Let a permutation be sortable by two stacks in series under the restriction that the permutation on the first stack avoids the classical pattern $1324$. Let $\operatorname{Av}(P)$…

Let $\mathfrak S_n$ be the set of permutations of length $n$, and let $\text{SC}_{1\underline{23}}$ and $\text{SC}_{3\underline{21}}$ be the corresponding vincular-pattern-avoiding…

Let $\mathfrak S_n$ be the set of permutations of length $n$, and let $\text{SC}_{1\underline{23}}$ and $\text{SC}_{3\underline{21}}$ be the corresponding vincular-pattern-avoiding…

Let $\mathfrak S_n$ be the set of permutations of length $n$, and let $\text{SC}_{1\underline{23}}$ be the vincular-pattern-avoiding stack-sorting map. Second-largest-preimage conj…

Let $\mathfrak S_n$ be the set of permutations of length $n$, and let $\text{SC}_{1\underline{23}}$ be the vincular-pattern-avoiding stack-sorting map. Unique-maximizer conjecture.…

Let $\text{SC}_{312}$, $\text{SC}_{\underline{31}2}$, and $\text{SC}_{3\underline{12}}$ be the vincular-pattern-avoiding stack-sorting maps, and let $\text{Sort}_n(f)$ denote the p…

For a permutation under West's stack-sorting map, its fertility number is the number of preimages under that map. Defant and Zheng's conjecture. The smallest fertility number congr…