266 problems

Let $r(x)$ be the generating function of the Rueppel sequence, and let the Hankel transform be reduced termwise modulo $2$. Modulo-2 periodicity conjecture. The Hankel transform of…

Let $r(x)$ be the generating function of the Rueppel sequence, and let the Hankel transform of a sequence be the sequence of determinants of its successive Hankel matrices. Periodi…

Let $n\geq 2$ and let $\Gamma$ be a subgroup of $\mathbf{SL}_n\mathbb{C}$. Let $\{\gamma_0,\ldots,\gamma_l\}$ be the equivalence classes of irreducible finite-dimensional complex r…

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 … and write its Stieltjes continued fraction as … Let $s_{b,c}(n)$ denote the corresponding Stieltjes parameters. Generalized Rueppel parameter conjecture. Then … where $P(k)$…

Let $r(x)$ be the generating function of the Rueppel sequence, and let the Hankel transform be the sequence of determinants of successive Hankel matrices. A005811 Hankel-transform…

Let $r(x)$ be the generating function of the Rueppel sequence, and let the Hankel transform be the sequence of determinants of successive Hankel matrices. A005811 Hankel-transform…

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…

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

Generalized Fibonacci analogue conjecture. The generating function

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…

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 $m\ge2$,

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…

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…

Let $K_{m,n}$ be the complete bipartite graph with part sizes $m$ and $n$, and let $P_{\textsc{Col},K_{m,n}}(1)$ denote the number of positions in Col (equivalently, by the biparti…

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 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…

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…

Let $\mathcal L$ be the Fibonacci Lie algebra, let $\lambda=(1+\sqrt 5)/2$, and let $\mathbf E(\mathcal L,t)$ denote its Euler characteristic. Euler-characteristic asymptotic conje…

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…