1,386 problems

Let $(s_n)$ be an appropriate sequence of positive integers, and let $G_n$ be the action graph associated with $s_n$. For $n\geq 1$, write $z_n$ for the number of new vertices labe…

The q-log-convexity conjecture. (i) The sequence $(\sum_{k=0}^n\binom nk^3q^k)_{n\geq8}$ is $q$-log-convex; for every integer $m\geq2$, the sequence…

Let $f_{CS}(x)$ be the generating function for the cyclic shift strategy of length $n$, and let $f_S(x)$ be the generating function for a deranged strategy $S$, meaning a strategy…

For each integer $m\geq 4$, let $f_m$ and $h_m$ be the polynomials defined in the paper, and let the ULC inequality refer to the ultra-log-concavity inequality for their coefficien…

Negative-coefficient conjecture. In the remaining 33 pairs of the polynomials $A_{abcd}$ and $B_{2-a,2-b,2-c,2-d}$, including the case where $p\neq 2$ for $a=b=c=d=1$, one of the t…

Let $\pow\geq2$, $k\geq1$, and let $a_1,\ldots,a_k,b$ be positive integers. Write $[r]_q:=1+q+\dots+q^{r-1}$ and, for an integer $\pow$, write…

Sub-leading Laurent coefficient conjecture. For every $d\ge5$ and every $m\ge2$,

Let $a_s(r,n)$ denote the number of two-toned tilings of a $1\times(n+r+s)$ grid with $r$ red squares, where the last $s$ tiles must be white. For all $s,r,n\geq 1$, the conjecture…

Let $C$ be a set, let $\boldsymbol{a}$ be a transitive $n\times n$ matrix with entries in $C$, and let $\mathbf r=\{r^{(c)}\}_{c\in C}\subset \mathfrak{g}\otimes\mathfrak{g}$ be a…

Let $xi_{\ell,r,c}$ denote the integer in the reduced $Gr_2$-simplex in layer $\ell$, row $r$, and NE-SW diagonal $c$, and set $\xi_{\ell,r,c}=0$ when there is no entry in that pos…

Arrange the entries in each layer of the $Gr_2$-simplex as coefficients of a polynomial in $a,b,c$ according to the arrangement described in the paper. Divisibility conjecture. Eac…

Recurrence and coefficient conjecture. The linear recurrence corresponding to $k$ has order $\left\lfloor k/2\right\rfloor+3$, and each coefficient $c_j(q)$ is a linear polynomial…

For each $n\geq 2$, let $\mathcal{G}_n$ be the set of Gilbreath sequences of length $n$, let $K_S$ denote the valid-extension set associated with $S$, and let $U_n$ be the doubling…

Coprime dice relabeling conjecture. There are no ways to relabel a die of size $r$ and a die of size $s$ without changing the frequencies of their sums.

Standard-Vandermonde determinant formula. The first appearance value is

Standard-Vandermonde value conjecture. For $n \geq 2$,

Unified row-shifted second-power lattice conjecture. For $n \geq 2$,

Multiplication-table lattice value conjecture. For $n \geq 2$,

Harmonic APD relation. For $n \geq 1$,

Hilbert-matrix first appearance value conjecture. For $n \geq 2$,

Row-shifted second-power lattice value conjecture. For $n \geq 2$,

Row-shifted second-power lattice degree conjecture. For $n \geq 2$ and any positive integer $d$,

Standard-circulant first appearance value conjecture. For $n \geq 2$,

Standard-circulant first appearance degree conjecture. For $n \geq 2$,

A hull configuration is a point configuration lying either on a line segment or on the boundary of a convex polygon; a blank side is an edge of the convex hull containing only two…