1,024 problems

An alternating sign matrix (ASM) of order $n$ is an $n\times n$ matrix whose entries are in $\{0,1,-1\}$ and whose nonzero entries in each row and column alternate, beginning and e…

Let $n(g)$ denote the number of numerical semigroups of genus $g$. Bras-Amorós' conjecture. For each integer $g\geqslant 1$, … This is a conjecture about the growth of the number o…

For integers $m\geq 1$ and $1\leq r\leq m-1$, let $d53c^{(m,r)}_n$ be the set of greedy $(m,r)$-Tamari intervals of size $n$, and let … be their generating function with the weight…

For a positive integer $n$, a tuple ${\bf r}=(r_1,\ldots,r_m)$ of nonnegative integers, and the polynomial ${\sf{M}}^n_{\bf r}(q)$ defined in the paper, the claim concerns its coef…

Product formula conjecture. There exists an integer sequence $\{o_n\}_{n \geq 0}$ such that

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…

Dyck-path dimension conjecture. The dimension of $DHA_n$ is given by the number of Dyck paths in ${\cal L}_n$.

Let $\sigma=\sigma_1\sigma_2\ldots\sigma_n\in\mathfrak{S}_n$. Write $\operatorname{Exc}(\sigma)=\{i: \sigma_i>i\}$ for the excedance set, let…

Let $\mathcal{M}_n(P_1)$ be the set of Stoimenow matchings avoiding $P_1$, let $\mathcal{P}_n{\bf (3+1)}$ be the corresponding class of $(3+1)$-avoiding Fishburn posets, and let…

For a matroid $\mathrm{M}$ of rank $r$, let $W_i$ denote the number of flats of rank $i$. Dowling–Wilson's top-heavy conjecture. One has … and $W_i\leq W_{r-i}$ for $i\leq r/2$. Th…

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

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

Complete-graph extremal conjecture. The number of arithmetical structures on $G$ is at most the number of arithmetical structures on the complete graph $K_n$:

Duncan–Steingrímsson's conjecture. On modified ascent sequences, these six patterns are all Wilf-equivalent, and the enumeration of modified ascent sequences avoiding any one of th…

Propp's conjecture. The $(a,b)$-benzel has

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 $G$ be an oligomorphic permutation group whose profile is bounded by a polynomial. Writing $varphi_G(n)$ for the number of $G$-orbits on $n$-element subsets, Cameron's conjectu…

Let $STS(n)$ denote the number of non-isomorphic Steiner triple systems on $n$ vertices, where $n$ is admissible, meaning $n\equiv 1,3\pmod{6}$. Wilson's conjecture. … This gives t…

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…

Di Francesco's conjecture. The number of domino tilings of the Aztec triangle is given by a product formula similar to the product formula for alternating sign matrices.

Let $f(n)$ denote the number of non-isomorphic matroids on an $n$-element set, where $n$ and $m$ are nonnegative integers. Welsh's conjecture. … The source presents this as a famou…

Let $\mathsf{PF}_n$ be the set of parking functions of length $n$, and define the sum statistic by $\mathsf{sum}(\pi)=a_1+a_2+\ldots+a_n$ for…

Niederreiter's conjecture. The number of $m$-dimensional $\alpha$-splitting subspaces is

Let $B_n$ be the number of non-isomorphic pseudoline arrangements of order $n$, and let $b_n:=\log_2(B_n)$. Knuth's upper-bound conjecture. … This conjecture concerns the asymptoti…

Let $\{x_n\}_{n\geq 0}$ be a sequence of nonnegative numbers, and define $z_n=\sum_{k=0}^n A_{n,k}x_k$ using the Eulerian triangle $\{A_{n,k}\}_{0\leq k\leq n}$. A sequence…