541 problems

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 … 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…

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

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 $f_1,\ldots,f_m$ 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…

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…

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 $T_{n\times m}(s,k)$ 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 $s\times s$.…

Associated Hermite matching-generating-function conjecture. The integral

…

Let $F(z)\in\mathbb{C}[[z]]$ and let $\alpha\in\mathbb{C}$. The following conditions are equivalent:nn1. $F$ is Möbius-covariant of weight $\alpha$, namelyn … n2. For every nonnega…

Let $n$ be a positive integer. A cyclic V-shaped permutation of $[n]$ is a permutation with a unique local minimum that is also a cycle, and let the minimum occur at position $k$.…

Growth conjecture. For $n \geq 12$,

For each positive genus $g$, let $S_{n,k}^{(g)}$ be the coefficients indexed by positive integers $n$ and $k$, and let $\chi^{(g)}$ be a triangular array of constants. Let the expr…

Let $S_{n,k}^{(g)}$ denote the coefficients indexed by $n$, $k$, and genus $g$, and let the expression referred to as … , with $g=3$, gives $S_{n,k}^{(3)}$ for all $n$ and $k$. The…

For each $n\in\mathbb{N}$, let $W_n$ be the set of permutations of size $n$, let $\operatorname{Simp}_n$ be the set of blockwise simple permutations, and define … where $S_n$ is th…

Generalized product formula conjecture. There exists a sequence of polynomials $o_n(k,t)\in\mathbb{Z}[k,t]$ such that

Let $\phi(n,d)$ denote the polynomial that gives the dimension of the space of quadrics in $n$ variables passing through $d$ general points and tangent to $\binom{n+1}{2}-d-1$ gene…

For positive integers $k,n,r$, let $N_{k,n,r}$ denote the number of cluster variables of rank $r$ in $\mathbb{C}[\operatorname{Gr}(k,n)]$. Cluster-variable counting conjecture. The…

Let $n\in\mathbb{P}$, and let $\mu(u,v)$ denote the Möbius function of the Bruhat-order interval $[u,v]$ in the poset of signed Wachs permutations $\operatorname{\mathcal{W}}(B_n)$…

Symmetry conjecture. For all integers $n$ and $k$ satisfying $1\leq k\leq n$, we have

Existence conjecture. For every positive integer $x$ whose length belongs to an admissible residue class modulo $144$, there exists a 3-inflatable permutation of length $x$.

Let $g_{n,k}$ denote the number of curve diagrams of genus $n$ and complexity $k$, as in the paper. Let $n\geq 2$ be an integer. Polynomial-growth conjecture. There exist positive…