157 problems

For each $n$, let $a_{n,k}$ be the number of permutations in ${\mathfrak{S}}_n$ whose longest increasing subsequence has length $k$. A sequence $(a_1,a_2,\dots,a_N)$ of positive re…

For a positive integer $n$, let $q$ be the residue-field cardinality and let $s$ be a complex variable. Let $\mathcal{Z}_{n,n,\mathfrak{o}}^{\mathsf{C}}(s)$ denote the symplectic l…

Let $\sigma=\sigma_1\sigma_2\ldots\sigma_n\in\mathfrak{S}_n$. Write $\operatorname{Exc}(\sigma)=\{i: \sigma_i>i\}$ for the excedance set, let…

Let $\mathfrak{S}_n$ be the symmetric group of permutations of $[n]$, and let $r\geq 1$. The statistics $r\mathsf{des}$, $r\mathsf{maj}$, $r\mathsf{exc}$, $r\mathsf{den}$,…

For each positive integer $n$, let $b_{n,k}$ denote the number of involutions in ${\mathfrak{S}}_n$ whose longest decreasing subsequence has length $k$, for $1\leq k\leq n$. A fini…

Gessel–Zhuang's conjecture. Every shuffle-compatible statistic is a descent statistic.

For a permutation $\tau$, let $\rm udr$, $\rm pk$, and $\rm des$ denote the permutation statistics of the length of the longest up-down run, the number of peaks, and the number of…

Let $\sigma$ be a uniformly random permutation in $S_n$, and let $\operatorname{as}_{n,k}$ denote the length of its longest $k$-alternating subsequence, where $1\leq k\leq n-1$. Ar…

Let $S_n$ be the set of permutations of $\\{1,2,\ldots,n\\}$, and let $\operatorname{des}_2$, $\operatorname{cyc}$, $\operatorname{pex}$, $\operatorname{des}$, and…

For each $n\geq 1$, let $P_n(x)=\sum_k P_{n,k}x^k$, where $P_{n,k}$ is the number of permutations of $[n]=\{1,2,\ldots,n\}$ whose longest increasing subsequence has length $k$. A p…

LSB Euler–Mahonian conjecture. The statistics \mathop{\text{\text{\rm cmaj}\text{LSB}}} and \mathop{\text{\text{\rm cinv}\text{LSB}}} are Euler–Mahonian on ordered partitio…

Euler–Mahonian pairing conjecture. Each of the eight statistics

Let $d_k(n)$ and $d_k^*(n)$ denote the two permutation statistics defined in the paper, and let $D_n$ denote the number of derangements (permutations without fixed points) in the s…

Nonnegative decomposition conjecture. For $n\geq 9$ and $k\geq 1$, the coefficients $b_{2n,k}$ are nonnegative integers.

Nonnegative decomposition conjecture. For $n\geq 1$ and $k\geq 0$, the coefficients $a_{n,k}$ are nonnegative integers.

The conjecture. For all $n\geq 4$:

Let $\mathcal P$ be the family … For $\Pi\in\mathcal P$, let $\mathfrak S_n(\Pi)$ denote the permutations of length $n$ avoiding every pattern in $\Pi$, and let…

Bivariate combinatorial interpretation. For all $n\ge1$ and $0\le k\le n$:

The layered-permutation conjecture. The maximal value of $athfrak{S}_w(1^n)$ for $w\in S_n$ is achieved when $w$ is a layered permutation.

Let $\mathfrak{S}_n$ be the set of permutations of $[n]$, and let $T_1,T_2,T_3$ be the statistics defined by … … … and let $S_{17}(p)$ be the largest $i$ such that $1,\dots,i$ appe…

Let $\tau$ be an arbor of size $n$, let $\mathcal Q_\tau$ be its associated arbor polytope, and let $\mathcal D(v)$ denote the union of the blocks at the descendants of a vertex…

Let $\mathfrak{S}_M$ be the set of multipermutations of the multiset $M$, and let $r\geq 1$. Let $\mathsf{des}_r$, $\mathsf{maj}_r$, $\mathsf{exc}_r$, and $\mathsf{den}_r$ be the c…

Let $G_{c,n}$ be the group of colored derangements, and let $\widetilde{inv}$ and $fmaj$ denote the statistics on $G_{c,n}$. For indeterminates $q$ and $t$, Symmetry conjecture. Th…

Classification conjecture. Up to equivalence, the only bicompatible permutation statistics are the descent set $\operatorname{Des}$, the peak set $\operatorname{Pk}$, the valley se…

Classification conjecture. The only nontrivial permutation statistics that are both shuffle-compatible and substring-compatible are the descent set, the peak set, and the valley se…