66,809 problems

Let $a_{11},a_{22},a_{33},a_{12},a_{13},a_{23},d$ be integers, and consider the generalized Markoff equation … Let $D$ and $A_{11},A_{22},A_{33}$ denote the quantities defined earl…

Let $G$ be a decorated unicyclic graph with cycle length parameter $g$, let $R_G$ be its curvature operator, and let $w^\ast$ denote the Perron metric. Write … and let $\varphi$ be…

Let $S$ be a subtraction set, and consider any two tie-breaking conventions in the associated self-interest cumulative subtraction game. For a heap size $h$, let the discrepancy be…

Let $d\ge 1$. Suppose $D(x)$ has finite support $\mathcal{U}$. If $D(x)$ is uniformly distributed on $\mathcal{U}$, and if every $x\in \mathcal{U}$ is on the boundary of…

Prasad and Ram's conjecture. For any $T\in M_{3m}(\mathbb{F}_q)$,

The universal-enveloping-algebra expression conjecture. For every $n$,

Regular-graph self-improvement conjecture. Any $d$-regular graph satisfying $\mathrm{CD}(0,\infty)$ satisfies $\mathrm{CD}(0,d)$.

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…

Let $G_n$ be the partition graph, let $\mathrm{Ax}_n$ and $\mathrm{Sp}_n$ denote its axial and spinal vertex sets, and let…

Central-quotient eigengap conjecture. Either $k_{>1}=|G|-1$, or

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 $a_{t,k}(n)$ count the hooks of length $k$ in all the $t$-core partitions of $n$. Hook-length bias conjecture for 5-core partitions. For all $n\geq 0$, one has … The inequaliti…

Let $G$ be a bipartite graph with independent sets $C_1$ and $C_2$, with $|C_1|=c_1\leq c_2=|C_2|$, as in Theorem. Let ${\mathcal F}_k(G)$ be the $k$-supertoken graph of $G$, and s…

Let $ped(n)$ denote the number of partitions of $n$ in which even parts are distinct and odd parts are unrestricted. Let $p\geq5$ be a prime satisfying … and let $t$ be a positive…

Product conjecture. The product of the $u_i$-tilted matrices

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 $p\geq 3$ be prime. A Hartke $\mathbb{Z}_p$-magic graph is a graph with the Hartke magic-labeling property over $\mathbb{Z}_p$; its order is its number of vertices. Low–Roberts…

Planar-graph unimodality conjecture. For any planar graph $G$ of order $n$, the sequence $\{\st{G}{k}\}_{k=1}^n$ is unimodal. This property is known for paths, cycles, trees, lolli…

Functional equation conjecture. The exponential generating function satisfies

Let $M_{m,n}(\mathbb{C})$ denote the space of complex $m\times n$ matrices. For a matrix, write $\|\cdot\|_F$ for the Frobenius norm, $\|\cdot\|_2$ for the spectral norm, and…

Let $\mathcal{C}$ be a set of columns of a dyadic-truncated Walsh--Hadamard matrix $W_\Phi$ with exactly one node, such that each branch is complete above the node. Let…

Let $\mathbb{K}$ be a finite field of characteristic different from $2$, and let $\mathbb{K}^{*}$ denote its multiplicative group. An element $x$ is a generator if it generates…

Let $calpha>2$, and for each $n$ consider the coefficient sequence $(c_{n,k})_{k\in\{1,2,\ldots,n\}}$. Unimodality conjecture. For every choice of $calpha>2$, the sequence … is uni…

For fixed $N$, let $B_{N-1,K}$ and $W_N^{\rm opt}$ denote the two-branch and standard dyadic Walsh--Hadamard truncation matrices described in the paper, with $K=0,\dots,N$. Two-bra…