266 problems

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…

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…

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…

Factor-occurrence rationality conjecture. For every fixed $k \ge 3$ and every non-empty factor $B \sqsubseteq \mathbf{W}^{(k)}$, there exist an integer $s(B) \ge 1$ and a polynomia…

Fix an integer $k\geq 1$. Let $p_{k,1}(2,n)$ denote the number of partitions of the rectangle $2\times n$ using the block-size restrictions indexed by $k$ and $1$ in the paper. A f…

Conditional positivity conjecture. $Q_{01}$ has nonnegative coefficients if and only if $p=2$ and $q\equiv 1 \mod 4$.

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…

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

Denominator conjecture. For all $k$, the rational generating function $F_k(x)$ can be written as

Fix $r\geq 2$ and $k\geq 2$, and let $v_r^{(k)}(n,t)$ be the sum of the $r$th powers of the coefficients of … with … Denominator-shape conjecture. There is an integer $m\geq 1$, de…

Fix $k\geq 2$ and let $v_3^{(k)}(n,t)$ be the sum of the cubes of the coefficients of … Define … Third-moment generating-function conjecture. One has … where … This conjecture give…

Generalized Fibonacci analogue conjecture. The generating function

Fix $k\geq 2$, $m\geq 1$, and $0\leq a<m$. Set … Let $h_{m,a}^{(k)}(n)$ be the number of coefficients of $I_n^{(k)}(x)$ congruent to $a$ modulo $m$, and define … Congruence-count r…

Fix $k\geq 2$ and define $H_{m,a}^{(k)}(x)$ as the generating function counting coefficients of $\prod_{i=1}^n(1+x^{F_{i+k-1}^{(k)}})$ that are congruent to $a$ modulo $m$. Parity-…

Let $C(x)=\sum_{n\geq 0}C_nx^n$ be the Catalan generating function, let $F(x,r)=C(x)^r$, and write $H_n(F(x,r))$ for the $n\times n$ Hankel determinant of the coefficients of…

Terwilliger systems' rank-generating polynomial factorization conjecture. There exist positive integers $d_1,d_2,\ldots,d_n$ whose sum is $d$ such that

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

Let $a_n$ be the number of triangulations of a fixed convex $n$-gon such that at least six triangles meet at each internal vertex. Let…

Let … where $i$ is an integer, and let $s$ be the unique formal power series in $x$ and $p$ satisfying … For the Hurwitz generating series $H_g(x,p)$, and for a partition $\theta$…

Minimal-period conjecture. The minimum period of $c_j(k)$ is $p_j$ for every $0\leq j\leq d$. This conjecture predicts that the divisibility data of the $a_i$ determine the exact p…

A pin word is a word encoding a permutation through the pin sequence construction; the set of permutations corresponding to pin words forms a permutation class. Rational generating…

Let $\mathcal{P}$ be the set of partitions, and define the partition functions … Here $f^\lambda$ is the dimension of the symmetric-group module indexed by $\lambda$. A singular po…

Let $M$ be a finite matrix defining a grid class $\operatorname{Grid}(M)$, and let $G(M)$ be the bipartite graph whose vertices represent the rows and columns of $M$, with an edge…

Let $\CSPP{n}^{k}$ be the restricted class of cyclically symmetric plane partitions, let $\overline U_r(c)$ be its statistic for $1\leq r\leq n$, and define … where $\MT{n}^{k}$ is…