347 problems

Let $w$ be a permutation, and let the Schubert polynomial corresponding to $w$ be considered for factorization into elementary symmetric polynomials. Pattern-avoidance conjecture.…

Let $\mathcal{M}_n(P_1)$ be the set of Stoimenow matchings avoiding $P_1$, let $\mathcal{P}_n{\bf (3+1)}$ be the corresponding class of $(3+1)$-avoiding Fishburn posets, and let…

Inversion monotonicity conjecture. Every pattern, except for the identity patterns, is inversion monotone. In particular, the pattern $1324$ is inversion monotone.

Archer et al.'s conjectures. For the indicated values of $n$, the following enumerations hold:

Let $P$ and $Q$ be patterns, and let $P\otimes Q$ denote their Kronecker product. Write $\operatorname{sat}(n,P)$ for the square saturation function and $\operatorname{ssat}(n,Q)$…

Strong Wilf-equivalence conjecture. Two binary words $p$ and $q$ are strongly Wilf-equivalent if and only if they are trivially equivalent.

Duncan–Steingrímsson's conjecture. On modified ascent sequences, these six patterns are all Wilf-equivalent, and the enumeration of modified ascent sequences avoiding any one of th…

Let $\mathfrak{S}_m$ denote the symmetric group on $m$ letters, and let $\pi,\tau\in\mathfrak{S}_m$. Write $\pi\sim\tau$ for ordinary Wilf-equivalence and…

Arratia's conjecture. For every permutation $\beta$ of length $k$,

A permutation matrix is a square $0$-$1$ matrix with exactly one $1$ in each row and column. The saturation function $\operatorname{sat}(n,P)$ is the minimum number of $1$ entries…

Let $A_n(\sigma)$ be the set of permutations of $[n]$ that avoid the pattern $\sigma$ and are alternating. Lewis's conjecture. For $n\geq 0$ and…

Symmetry conjecture. For all $m \ge 1$ and all $l_1, \ldots, l_m \ge 1$, the generating functions for the even involutions in $I(3412, [l_1, \ldots, l_m])$ and for the odd involuti…

For each positive integer $n$, let $\operatorname{av}_n^k(1324)$ be the number of 1324-avoiding permutations of length $n$ with $k$ inversions. Unimodality conjecture. The sequence…

Billey–Crites conjecture. The variety $X_w$ is smooth if and only if $w$ avoids the patterns $3412$ and $4231$.

Fine-sequence conjecture. For every $n\geq 1$,

Grassmannian enumeration conjecture. For every $n\geq 1$,

Pattern-avoidance conjecture for lci-ness. Non-lci-ness in general can be characterized by pattern avoidance.

Let $S_n$ be the symmetric group, let $0<k<n$ be an integer, let $v in Sn$ avoid the pattern $12cdots(k+1)$, and let $operatorname{Imm}v X$ denote the corresponding Kazhdan–Lusztig…

Alternating Möbius function conjecture. The Möbius function is alternating, meaning that it is

Rank-3 maximum conjecture. The maximum absolute value of the Möbius function on intervals of rank $3$ is

Enumeration conjecture. Each class $\mathcal{U}_{2k+1}(\tau^{(1)},\tau^{(2)})$ is counted by the corresponding OEIS sequence in the table: $A001764$ for $(312,3421)$, $A122368$ for…

The Type B non-decreasing parking functions are the elements of the submonoid of $\operatorname{NDPF}_{2N}$ generated by $\mu_i=\pi_i\pi_{2N-i}$ for $1\leq i\leq N$, with…

Let $ath]σ\in\mathcal S_m$ be a non-overlapping pattern, meaning that two occurrences cannot overlap in more than one position. Let $\alpha_n(σ)$ be the number of permutations of l…

Let $k\geq 1$ and let $\tau$ be a pattern. For patterns, write $\pi\stackrel{I}{\sim}\sigma$ when they are equally restrictive for involutions under pattern avoidance. Jaggard's co…

Let $Y$ be the Young diagram under consideration, let $S_Y$ denote its transversals, and let $S_Y(213)$ denote the transversals avoiding the pattern $(213)$. For a transversal…