552 problems

Stanley's conjecture. For $n\geq4$ and $n\geq5$, respectively,

The in-degree enumeration conjecture.

Let $n$ be a positive integer and let $j$ be a positive integer. A lucky spot is a parking spot occupied by a car whose preferred spot is that spot. Let $f_j(n)$ denote the polynom…

For each integer $n\ge 3$, let $\vec K_n$ denote the complete directed graph on $n$ vertices, and let $\genfrac{\{} {\}}{0pt}1{\vec K_n}{n-2}$ denote the number of admissible parti…

For a fixed integer $k\geq 1$, let $P'(2,n,k)$ denote the number of prime 2-ball, $k$-hand passing patterns of length $n$. As $n$ tends to infinity, Exponential lower-bound conject…

Let $n\geq 1$ and $k\in \{0,1,\dots,n(n-1)/2\}$. The quantity $\operatorname{upf}_{n,k}^{\operatorname{inv}}$ denotes the number of unit interval parking functions of length $n$ wi…

Let $\mathfrak{M}_n$ be the set of minimally-sorted permutations under $s_{123,132}$. Enumerative conjecture. For every positive integer $n$, … This gives a recurrence between the…

For integers $a,b\geq 0$, let $\mathsf{Shuf}(a,b)$ be the shuffle poset, with rank function $\mathsf{rk}$, and define … For $n>0$, consider the case $a=n-1$ and $b=1$. The interval…

For each $k$, let $\mathsf{M}_k^{{||}}$ and $\mathsf{M}_k^{{\ddagger}}$ denote the parallel and series transition matrices, and let $\mathrm{Eul}_{k}(t)$ be the Eulerian polynomial…

For each $k$, let $N_k$ be the matrix appearing in the expression for the generating series $\mathcal{R}_k(t)$, and write $\pi_{N_k}(t)$ and $\chi_{N_k}(t)$ for its minimal and cha…

Transposed Catalan triangle formula. The number of such permutations is

For each integer $k\geqslant 2$, let $\gamma_k$ be the constant in the asymptotic growth … of the maximal ranks of at-most-$k$-furcating rooted trees with $n$ leaves. Monotonicity…

Let $a_n$ denote the number of Motzkin paths of length $n$ satisfying the rule specified in the source. For $n\geq 1$, let A005802 and A216947 denote the cited OEIS sequences, with…

Functional equation conjecture. The exponential generating function satisfies

Let $T$ be an $m\times n$ table with $2n\geqslant m$. Let $\mathcal{C}(T)$ be the coefficient matrix whose entries are the coefficients arising when the relevant quantities…

Let … be the multivariable exponential generating function for submonoids of products of finite commutative monoids. Rationality conjecture. The generating function $G(x,y,z)$ is a…

Let $D(r,c,d)$ denote the number of ways to place $d$ non-bonding dominoes on an $r\times c$ rectangular board. Quadratic formula conjecture. For integers $r,c\ge 3$, … The formula…

Let $\alpha_{n,m}$ denote the number of optimal solutions for the $n\times m$ grid in the paper's notation. In the case of a grid with $2$ rows and $2k+1$ columns, where $k\geq 0$,…

Let $([d],\mathcal{L})$ be a covering clutter, let $s$ be its magic subspace, and let a magic permutation be a permutation of $[d]$ realizable by a positive point in $s$. For a per…

Associated Hermite matching-generating-function conjecture. The integral

Let $m,d,n$ be nonnegative integers, and let $4\lambda\vdash m$ be a partition of $m$ of length $k$. For each $i$, let $4\mu_i(\lambda)$ be the number of parts of $4\lambda$ of siz…

Linear cumulant conjecture. Every cumulant is a linear function of $n$ for all $k\geq 1$.

Let $T_{n\times m}(s,k)$ denote the number of ways to tile an $n\times m$ rectangle using $s\times s$ squares and $1\times1$ squares, with exactly $k$ squares of size $s\times s$.…

For a perfect matching $M$ on $[2m]$, let ${\rm cr}(M)$ and ${\rm ne}(M)$ denote its numbers of crossings and nestings, respectively, and let ${\mathcal{M}}_{2m}$ be the set of per…

Let $f_1,\ldots,f_m$ be positive linear forms on $\mathbb{R}^d$, and let $s$ be the subspace on which they all vanish. For a permutation $\sigma$ of $[d]$, define the reverse domin…