211 problems

Let $n$ and $t$ be positive integers with $n>2t$, and let $\mathcal{M}_n(n,n)$ denote the set of perfect matchings of the complete bipartite graph with $n$ vertices in each part, v…

For a permutation $w$, let $\mathrm{S}_{w}$ denote the Stanley symmetric function. A symmetric function is Schur positive if its expansion in the Schur-function basis has only nonn…

Let $P_n$ be the path on $n$ vertices, and let $F(G)$ denote the maximum size of a family of permutations of the vertices of a graph $G$ such that every two permutations have, in s…

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

Two disjoint order-isomorphic subsequences of a permutation are called twins, and let $\tau(n)$ denote the maximum length of twins contained in every permutation of length $n$. Gaw…

Filz's prime-circle conjecture. For every positive even integer $n$, there exists a permutation $\tau\in S_n$ such that all of these cyclic adjacent sums are prime.

Let $n$ be a positive integer. A three-term arithmetic progression $(a,a+r,a+2r)$ in $\mathbb{Z}/n\mathbb{Z}$, with not all terms equal, is preserved by a permutation…

Let $n$ and $k$ be integers with $1leq k<n$. For $i,j\in\mathbb{Z}_n$, let … be the circular distance, and let $\sigma(n,k)$ be the largest positive integer $s$ for which there is…

Lazar and Wachs' cycle-distribution conjecture. The number of D-permutations on $[2n]$ with $k$ cycles equals the number of E-permutations on $[2n]$ with $k$ cycles for all $k$. Co…

Ashlock–Tillotson conjecture. The minimum length of an $n$-superpermutation is

Let $n$ be the size of a random permutation, and let $T_{n,p}$ denote the number of strong interval trees with $n$ leaves and $p$ prime vertices. Define … where $T_n$ is the total…

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…

Steingrímsson–Williams joint distribution conjecture. The joint distribution of tableaux according to the number of rows and the number of essential 1s equals the joint distributio…

Steingrímsson–Williams conjecture. The distribution of permutation tableaux according to the number of essential 1s is equal to the distribution of permutations according to the nu…

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…

Let $\pi$ be a permutation, and let $L_\pi$ be the linear map associated with $\pi$ whose surjectivity is equivalent to the locally factorial property of the corresponding Schubert…

Brändén–Mansour conjecture. The number of words over the ordered alphabet $[n]$ of length $n$ that avoid $\pi$ is at most exponential in $n$; equivalently, for each permutation…

Let $S_n$ be the symmetric group, and let a displacement pattern be a tuple $\alpha=(a_1,a_2,\dots,a_n)$ recording the displacements of a permutation, with the entries distinct. Th…

The conjectured formula for $\mathfrak{D}_w$. For special $w$,

Let $n>2$ be prime, and define a permutation $g$ of $\mathbb Z_n$ by … Order permutations lexicographically, and let $\Psi(n)$ be the minimum number of collinear triples in a permu…

Let $n>2$ be prime, and let $Psi(n)$ denote the minimum number of collinear triples in the graph of a permutation of $mathbb Zn$ in $mathbb Zntimesmathbb Zn$. The preceding bounds…

Existence conjecture. An $m$-symmetric permutation on sufficiently many symbols exists for every $m$, and one likely exists on $h(m)$ symbols.

Let $\mathcal{F}'$ be the modified oracle using the truncated CCL process with $l=\log N$, an oracle $\mathsf{RND}_3$ making $k$ calls to a permutation oracle, and parameters as de…

Monotonicity conjecture. The sequence

Equivalent form of Bebeacua's conjecture. For every positive integer $n$,