61 problems

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…

Hilmarsson's conjecture.

Dokos et al.'s conjecture. The patterns $3142$, $3241$, and $4132$ are $\operatorname{maj}$-Wilf equivalent, and the patterns $1423$, $2314$, and $2413$ are $\operatorname{maj}$-Wi…

Rearrangement Conjecture. If $u$ and $v$ are Wilf equivalent, then they are rearrangements of each other; that is, they have the same multiset of letters. The converse is false: fo…

Let $\mathcal{S}_n$ be the set of permutations of $[n]$, and let $\mathcal{S}_n(\tau)$ denote the permutations avoiding the pattern $\tau$. For a permutation $\pi$, its major index…

Let $A$ and $B$ be permutation matrices. For the block-diagonal permutation matrices … write $A\sim B$ when the corresponding permutation matrices are Wilf-equivalent. Block-transp…

There are $7^3=343$ triples of binary relations $(\rho_1,\rho_2,\rho_3)$, where each relation belongs to $\{<,>,\leq,\geq,=,\neq,-\}$. Two triples are equivalent if they have the s…

Burstein–Han–Kitaev–Zhang conjecture. The two displayed partially ordered patterns are shape-Wilf-equivalent:

Joint distribution equivalence of S21 and S22. The first pair of mesh patterns, with points at $(1,1),(2,2),(3,3)$ and at $(1,3),(2,2),(3,1)$, satisfies $\sim_{jd}$, and the corres…

Let $k\ge 2$. Consider the two partially ordered patterns (POPs) shown in the source: the first has a central element labeled $k$ below elements labeled $1,2,\ldots,k-1$, while the…

For a permutation $\tau$, let $\operatorname{Destop}(\tau)$ and $\operatorname{Desbot}(\tau)$ denote its descent top and descent bottom sets. Patterns are…

Let $S_4$ be the set of permutations of length $4$. For a permutation $\tau$, define its descent bottom set by … Two patterns are Desbot-Wilf equivalent if their avoidance classes…

Let $v$ and $w$ be permutation patterns. They are c-Wilf-equivalent in permutations if … for every $n\in{\mathbb N}$, and they are strongly c-Wilf-equivalent in permutations if … f…

Let $P$ be a set of indecomposable patterns. Let $\mathsf{LMAX}$ denote the set-valued statistic recording left-to-right maxima, and let $\mathsf{iar}$ and $\mathsf{comp}$ denote t…

Let $P$ be a pair of patterns of length $4$ other than $(2413,3142)$, $(2413,4213)$, and $(3412,4312)$. Here, $mathsf{iar}$ denotes the length of the initial ascending run, and…

Forest-structure-Wilf conjecture. If $S$ and $S'$ are forest-Wilf equivalent, then they are forest-structure-Wilf equivalent.

Forest-Wilf implication conjecture. If $S$ and $S'$ are forest-Wilf equivalent, then $S$ and $S'$ are Wilf equivalent with respect to pattern avoidance in permutations.

Forest-Wilf equivalence conjecture. The following three equivalences hold:

Let $m\geq3$ and $r\geq1$, and let $q$ be any classical pattern of size $m$. Increasing-pattern extremal conjecture. There exists $n_0\in\mathbb{N}$ such that for every natural num…

For a distant pattern, write a square between two consecutive letters to indicate the constrained gap at that position. Consider the classical patterns $1234$, $1243$, and $2143$,…

Let $\mathfrak{D}^1_{2n}$ denote the set of Dumont-1 permutations of length $2n$, and let $\mathfrak{D}^1_{2n}(\tau)$ denote those avoiding the pattern $\tau$. Burstein–Jones's Wil…

Let $\mathbf I_n(0012)$ denote the set of inversion sequences of length $n$ avoiding the pattern $0012$. Jun Ma and Lin's conjecture. For $n\geq 1$, we have … In other words, the u…

For a tuple of patterns, write $(\rho_1,\ldots,\rho_k)\thicksim(\sigma_1,\ldots,\sigma_\ell)$ when the corresponding inversion-sequence avoidance classes are Wilf-equivalent, meani…

Let $\mathbf I_n(\rho_1,\rho_2,\rho_3)$ denote the set of inversion sequences of length $n$ avoiding the indicated patterns or relations. Martinez and Savage's conjecture. For…

For $n\geq 1$, let $\operatorname{Av}_n(231)$ and $\operatorname{Av}_n(132)$ denote the sets of permutations in $S_n$ avoiding the patterns $231$ and $132$, respectively. Let…