452 problems

Let $\mathfrak{S}_n$ be the symmetric group on $[n]=\{1,\dots,n\}$. For a fixed permutation $\pi$, let $\operatorname{Av}_n(\pi)$ be the set of permutations in $\mathfrak{S}_n$ tha…

Let $\mathfrak{S}_n$ be the set of permutations of $[n]$, let $\mathcal{S}$ be the stack-sorting operation, and let $\mathfrak{S}_n^2$ denote the permutations sorted by applying…

Let $\rho$ be a finitely based permutation class, and let $f_\rho(x)$ be its generating function. Noonan–Zeilberger conjecture. Every finitely based permutation class has a D-finit…

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…

Let $P$ be a permutation matrix, and let $\operatorname{ex}(n,P)$ denote the extremal number of $P$. Füredi–Hajnal conjecture. For every permutation matrix $P$, … This conjecture i…

Gessel's conjecture. The polynomial $A_n(s,t)$ has the symmetry property of being gamma-positive: it can be expressed as a non-negative integer linear combination of the bivariate…

Let $w$ range over permutations in $S_n$, and let $\mathfrak S_w(x_1,\dots,x_n)$ denote the Schubert polynomial indexed by $w$. A permutation is layered if it is a concatenation of…

Stanton's unimodality conjecture. The coefficients of each polynomial $C_n(x)$ form a unimodal sequence.

Hankel total positivity conjecture. The sequences of polynomials

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

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

Elizalde–Noy conjecture. Among all consecutive patterns of a fixed length $m$, the increasing pattern $12\dots m$ and, by symmetry, the decreasing pattern $m\dots21$ should be the…

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 $k\geq 1$, let $f(k)$ be the minimum $n$ such that there exists a permutation $\sigma\in S_n$ containing every permutation in $S_k$ as a pattern; such a permutation is called a…

Let $S_n$ be the set of permutations of $[n]$. A permutation is $k$-universal if it contains every permutation in $S_k$, and choose a permutation uniformly at random from $S_n$. Al…

For $n,r\in\mathbb{N}$, let $\mathfrak{S}_n^r$ be the set of $r$-stack-sortable permutations and let $A(\mathfrak{S}_n^r;x)$ denote its descent polynomial. Bona's conjecture. The p…

Hilmarsson's conjecture.

Let $\mathcal{W}'_2(n)$ denote the set of objects defined in the paper for which the enumeration $|\mathcal{W}'_2(n)|$ is considered, and let the $n$th Motzkin number be the Motzki…

Large-order reverse-strategy conjecture. For every $k\geq 3$, there exists $n_0(k)$ such that the reverse strategy is a winning strategy for Player II on $S_n$ for all…

Let $L_k$ denote the minimum length of a permutation containing every permutation pattern of length $k$. Arratia's conjecture. Asymptotically, … Arratia's conjecture concerns the a…

Let $p$ be a permutation of $[n]$. A $k$-modal subsequence is a subsequence with at most $k$ changes in direction, where each change is between increasing and decreasing. Chung's c…

Stieltjes moment conjecture. The sequences of polynomials

Fibonacci enumeration conjecture. The sequence $(a_n)$ satisfies

Atkinson–Sack conjecture. The generating function for the enumeration of sortable permutations is rational for every value of $k$.