99 problems

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…

For $n\geq1$ and $0\leq k\leq\lfloor(n-1)/2\rfloor$, define … A symmetric polynomial with center of symmetry $(n+1)/2$ is $\gamma$-positive when its expansion in the corresponding…

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 $g\geq 1$ and $\ell\geq 1$, and let $g\mathsf{exc}_\ell$ and $g\mathsf{den}_\ell$ denote the $g$-gap $\ell$-level excedance number and $g$-gap $\ell$-level Denert statistic, re…

Equidistribution conjecture. The statistics $\operatorname{pix}$ and $\operatorname{fix}$ are equidistributed over $\mathfrak{S}_{n}(\Pi)$ for all $n\geq0$ if and only if $\Pi$ is…

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…

Asymptotic extremal covariance conjecture.

Strict inequality conjecture. The inequality in is strict for all $m>0$. This would imply that $\operatorname{Var}(X_\tau)$ has order $2m-1$ in $n$, so the standard deviation of…

Guo–Zeng conjecture. For every $n\in\mathbb{N}$, there exist coefficients $a_{n,i}\in\mathbb{N}$ such that

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…

Nonnegative decomposition conjecture. For $n\geq 9$ and $k\geq 1$, the coefficients $b_{2n,k}$ 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 $0\le k\le n$:

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 $G_{c,n}$ 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…

Adams et al.'s conjecture. The number of weak excedances that are also mid-points of a decreasing subsequence of length $3$ (Statistic $373$ in FindStat) is equidistributed with th…

Let a mesh pattern be specified by a set of marked points in a $3\times3$ grid together with a set of shaded cells; let $\sim_{jd}$ denote equality of joint distributions for the c…

Monotone total positivity conjecture. For all $i,j,r\in\mathbb{N}$, the determinant of this submatrix is a monotonically increasing function of $i\in\mathbb{N}$.

Fulman–Kim–Lee–Petersen conjecture. The polynomials $\sum_{i=0}^n P_{n,k}t^k$ and $\sum_{i=0}^n Q_{n,k}t^k$ are all real-rooted for $n\ge 5$. This is an open problem proposed as pa…

Short-distance inversion conjecture. For $n\geq 2$ even, the number of inversions of distance at most $3$ (Statistic $494$) exhibits the cyclic sieving phenomenon under involutions…

Weak-excedance mid-point conjecture. The number of weak excedances that are also mid-points of a decreasing subsequence of length $3$ (Statistic $373$) is equidistributed with the…

Coxeter-length difference conjecture. The difference in Coxeter length of a permutation and its image under the Simion–Schmidt map (Statistic $123$) exhibits the cyclic sieving phe…

A valleyless-tieless parking function of length $n$ is a parking function $(a_1,\ldots,a_n)$ with no index $1\leq i\leq n-1$ such that $a_i=a_{i+1}$ and no index $2\leq i\leq n-1$…