44 problems

Let $I_3=123$ and $J_3=321$. For a permutation pattern $\tau$, prefix exchange over alternating involutions means equality of the avoidance-class cardinalities after adjoining any…

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 the $13$ pattern sets be the sets referred to in Corollary, and let $\{13452,23451\}$ be represented by the partially ordered pattern (POP) shown in the source. Wilf-equivalenc…

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…

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…

Let ${\mathcal{C}}$ be a class of permutations, and let ${\mathcal{C}}_3$ denote its permutations of length three. Let ${\mathcal{S}}_3$ be the set of all permutations of length th…

Let $pi1 otinPik$ and $pi2 otinPil$ with $k otin l$. For a pattern $pi$, write $pi^c$ for its complement, and let $Pil(pi1)$ denote the set of length-$l$ patterns avoided jointly w…

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…

Let $cpi,\tau\binom{}{}$ be permutations in $cmathfrak{S}_m$, and call a permutation in standard form when $cpi_1<cpi_m$ and $cpi_1+cpi_m\boxtimes m+1$. Let $r_{n,k}^{cpi}$ denote…

Let $M\mathcal I_n(\pi;q)$ denote the major-index generating function over involutions in $\mathfrak S_n$ avoiding a pattern $\pi$, and let $[\pi]_{M\mathcal I}$ denote the corresp…

For a permutation pattern $\tau$, let $F_n^{\operatorname{des}}(\tau;q)$ denote the descent generating polynomial over permutations in $\operatorname{Av}_n(\tau)$, and for sets of…

For a permutation pattern $\tau$, let $F_n^{\operatorname{des}}(\tau;q)$ denote the descent generating polynomial over permutations in $\operatorname{Av}_n(\tau)$, and write…

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…

Harmse–Remmel conjecture. $P$ and $Q$ are c-Wilf equivalent if and only if they are strongly c-Wilf equivalent. The conjecture extends Nakamura's $k=1$ formulation to $k\geq2$; the…