222 problems

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…

Fix $k$ and let $\mathcal{R}_k$ be the state space of the finite $k$-chain. For a partition $\lambda\in\mathcal{R}_k$, let $\lambda^{+i}$ denote the state obtained by adding a box…

Lecture hall position conjecture. If $\pi=\varphi(\lambda)$, then

Density conjecture. The natural density of indices $m$ for which $p_0(m)$ is even exists and equals

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

For each positive integer $n$, let $\mathcal{O}(n)$ be the set of overpartitions of $n$, and define…

Let $F_1(q),F_2(q),F_3(q),F_4(q)$ denote the four weighted truncated pentagonal-number series appearing in Merca's conjectures. The conjectures assert that $[q^n]F_i(q)\ge 0$ for e…

Let $\mathcal{O}_{n,m}$ be the set of partitions $\lambda$ of $n$ whose parts are odd and whose largest part satisfies $\lambda_1\le m$, and let $\mathcal{D}_{n,m}$ be the set of p…

Prove the eighteen explicit conjectural sum–product identities arising in the modulo-nine Kanade–Russell program: ten identities obtained from reflections of finite forms under…

Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_\ell)$ range over partitions into distinct parts, and define its perimeter by $\operatorname{per}(\lambda)=\lambda_1+\ell$. Let…

Let $\mathcal{C}_{a,b,m}(q)$ denote the infinite copartition-product generating function of Burson and Eichhorn, and let $\mathcal{C}_{a,b,m;N,M}(q)$ denote its finite version. The…

Let $c\phi_{18}(n)$ denote the number of $18$-colored generalized Frobenius partitions of $n$. Prove that, for every integer $n\geq 0$, $c\phi_{18}(30n+19)\equiv 0\pmod{16}$ and…

Andrews–El Bachraoui positivity conjecture. For every positive integer $k$, the series $F_{k,1}(q)$ is positive.

A partition $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_\ell)$ is gap-free, or compact, when $0\leq\lambda_i-\lambda_{i+1}\leq 1$ for all $1\leq i\leq\ell-1$. Let $a_2(n)$ denote…

Let $\mathrm{den}(n,x)$ be the denominator polynomial associated with sums of reciprocals of subsum polynomials over partitions of $n$. Log-concavity conjecture. The sequence of co…

Merca's conjecture. This theta series has non-negative coefficients. The conjecture refines the established positivity theorem for averaged truncations of Jacobi's triple product i…

For nonnegative integers $A,B$, define the $q$-binomial coefficient by … Suppose that $M,N\in\mathbb Z^+$, that $\alpha$ and $\beta$ are positive rational numbers, and that $K$ is…

Let $t_o(n)$ be defined by … where $q$ is a complex number with $|q|<1$ and $(a;q)_n$ denotes the standard $q$-Pochhammer symbol. Andrews–El Bachraoui's conjecture. If $8n+9$ has a…

Andrews–Bachraoui conjecture. There holds

Monotonicity conjecture. Let $t\geq3$ be an integer. Then

Laguerre threshold conjecture. For $k=1$ or $2$ and $1\leq m\leq14$, the sequence $\Delta_k(n)$ satisfies the Laguerre inequality of order $m$, namely $L_m(\Delta_k(n))\geq0$, for…

Let $G(q)$ be the $q$-series … and let $e(n)$ and $o(n)$ denote respectively the numbers of partitions of $n$ into odd parts whose index is even and odd. Coll–Mayers–Mayers conject…

Keith–Zanello's self-similarity conjecture. For a positive proportion of primes $p$, one has

Let $n$ be a positive integer. A partition of $n$ is a weakly decreasing sequence of positive integers with size $n$ and length $\ell$. For $1\leq j\leq\ell$, define…

Eventual bias conjecture. There exists a constant $N_{a,b,m}>0$ such that