255 problems

Pattern-avoidance conjecture. The map $\mathrm{hat}_{\max}$ restricts to the following bijections:

Mallows unimodality conjecture. If $\pi$ is any consecutive permutation pattern, then the function $p\mapsto\mathbb{E}_{n,p}(\pi)$ is unimodal.

For every $n\ge 1$ and every subset $S\subseteq [n]=\{1,\ldots,n\}$, the numbers of permutations in the two avoidance classes having descent-top set $S$ are equal: … Here…

Let $b_{2n,n}$ denote the number of permutations of length $2n$ that contain an increasing subsequence of length $n$, and define $B(z)=\sum_{n\geq 0} b_{2n,n}z^n$. The conjecture i…

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Determine the exact value of the Stanley–Wilf limit $\operatorname{gr}(\operatorname{Av}(1324)):=\lim_{n\to\infty}\lvert\operatorname{Av}_n(1324)\rvert^{1/n}$, where…

Let $S_n$ be the set of permutations of length $n$, let $SC_{231}:S_n\to S_n$ be the stack-sorting map whose stack avoids consecutive occurrences of $231$, and let…

Let $\lambda$ be any partition and let $w\in S_n^{\lambda}$. A flagged Grothendieck polynomial is understood in the sense used in the paper, and $w$ is 312-avoiding when it has no s…

Hankel total positivity conjecture. The sequences of polynomials

Let $\rho$ be a permutation, and let $D(\rho)$ be the principal downset of all permutations contained in $\rho$, ordered by permutation-pattern containment and ranked by permutatio…

Gao's conjecture. For any permutation $w$, $c_w\geq 0$. Equivalently,

Let $v=12\dots k$ be the increasing permutation pattern, let $N_{12\dots k}$ count occurrences of this pattern in permutations, and let $\operatorname{ch}_n(RN_{12\dots k})$ denote…

For $n\geq 1$, define … where $\nu_w=\mathfrak{S}_w(x)|_{x_i=1}$. A layered permutation $w(b_1,\ldots,b_k)$ is the concatenation of decreasing blocks on consecutive intervals, wher…

Symmetry characterization for zig-zag classes. If two zig-zag classes are symmetrically equivalent, then one is formed by complementing the labels of the other or by reversing all…

Let $D4AE_n$ denote the symmetric group on $n$ letters. Let $D4AE_n(231)$ be the set of permutations avoiding the pattern $231$, and let $S_n(t)$ be its descent polynomial. Write ……

Let $S_n$ be the symmetric group on $n$ letters. For permutations $u,w\in S_n$, let $D_u^w$ denote the dual Schubert polynomial, and let SCNP denote the property that its Newton po…

Given a set of permutations $S$, let $\operatorname{Av}(S)$ denote the set of permutations avoiding the patterns in $S$. Let $\mathfrak{L}_{n,m}$ be the family of looms and let…

A bicrucial permutation is a square-free permutation such that prepending or appending any single element creates a square. The existence conjecture. There exist arbitrarily long b…

Let $w\in \mathfrak{S}_n$ be a $321$-avoiding and $2$-repeating permutation. Let $u,v\in\mathfrak{S}_n$ satisfy $u\leq v\leq w$, and let $\tilde{R}_{u,v}(t)$ denote the correspondi…

Upper-bound conjecture. The values $d_n$ are bounded from above by $0.6040$.

Let $k$ be a positive integer. For a permutation $\tau$ of $[n]=\{1,\dots,n\}$, let $m_k(\tau)$ denote the number of monotone subsequences of length $k+1$. Myers' asymptotic conjec…

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…

Elizalde–Noy maximality conjecture. The increasing pattern $\sigma$ is the maximal pattern, in the sense that

Growth conjecture. For $n \geq 12$,

Let $M$ be a cyclic partial multiplication matrix containing $\ell$ non-empty cells. A coil is a permutation represented by a cyclic sequence of points following the cycle of non-e…