14 problems

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

Let $s$ be a positive integer, let $\Phi$ be a root system of rank $n$ with Coxeter number $h$ and exponents $e_1,\ldots,e_n$, and let $X$ be the set of facets of the generalized c…

Cyclotomic Catalan conjecture.

Bivariate combinatorial interpretation. For all $n\ge1$ and $0\le k\le n$:

For positive integers $a,b$ and $k$, define … where $c_d(k)_q=\sum_{s\mid\gcd(d,k)}\mu(d/s)[s]_q$. Here $[n]_q$ is the $q$-integer, $\mu$ is the Möbius function, and…

Let $\mathcal{L}=\mathbb{Z}^{a-1}$ be the weight lattice, let $\operatorname{Box}[c'+1,c]$ be the indicated slab of the fundamental box, and order lattice points by the tilted part…

Let $\mathcal{L}=\mathbb{Z}^{a-1}$, let $\operatorname{Box}$ be the fundamental box, and for coprime $a,c$ let $c'$ be the largest integer less than $c$ with $\gcd(a,c')=1$. A stan…

For each integer $a\geq3$, let $\operatorname{Cat}((a;c))_q$ denote the $q$-Catalan germ associated with an integer $c$, as defined in the paper. q-Catalan germ positivity conjectu…

Let $\mathcal{L}=\mathbb{Z}^{a-1}$ be the weight lattice, let $\mathcal{R}$ be its root sublattice, and let $\mathsf{T}(x_1,\ldots,x_{a-1})=x_1+2x_2+\cdots+(a-1)x_{a-1}$. A Johnson…

Let $\mathcal{L}=\mathbb{Z}^{a-1}$ be the weight lattice, let … be the root lattice, and let $\mathsf{T}(x_1,\ldots,x_{a-1})=x_1+2x_2+\cdots+(a-1)x_{a-1}$. Johnson's conjecture. Th…

For coprime positive integers $a$ and $b$, define the rational $q$-Catalan number … Here $[n]_q=(1-q^n)/(1-q)$ and $\mathbb{N}[q]$ denotes polynomials in $q$ with non-negative inte…

For integers $m\geq 2$, let … denote the generalized $q$-Catalan numbers, and let their coefficients be understood in the same way as for the ordinary $q$-Catalan numbers. Generali…

Let $C_n(q)=\sum_{k=0}^{n(n-1)}m_n(k)q^k$ denote the $q$-Catalan numbers and let $m_n(k)$ be their coefficients. Log-concavity conjecture. There exists an integer $t$ such that, wh…

Let $C_n(q)=\sum_{k=0}^{n(n-1)}m_n(k)q^k$ denote the $q$-Catalan numbers and let $m_n(k)$ be their coefficients. Eventual unimodality conjecture. The sequence … is unimodal when…