215 problems

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

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…

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

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…

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

Let $k\geq 2$, let $\mathcal{R}_k$ be the state space of the finite $k$-chain, let $\pi$ be its stationary distribution, and set … For each $\lambda\in\mathcal{R}_k$, write…

Hirschhorn–Sellers conjecture. For all $n\geq 1$ with $3\nmid n$,

Extension conjecture. Corollary 12 also holds for primes $p>23$.

For any nonnegative integer $n$, define … Let $Q_2(n)$ denote the partition function in the source. Weaker $Q_2$ truncated-sum inequality. For $n\geqslant0$ and $k>0$, … The source…

For any integer $n$, define … Let $b_6(n)$ denote the $6$-regular partition function, and let $(\rho_j)_{j\geqslant0}$ be the coefficient sequence in the source. 6-regular partitio…

For any integer $n$, define … Let $b_6(n)$ denote the $6$-regular partition function. 6-regular partition inequality. For $n\geqslant0$ and $k>0$, … with strict inequality if…

For $n,k\geqslant0$, let $p(n)$ denote Euler's partition function, let $Q_2(n)$ denote the partition function used in the source, and let $(\rho_j)_{j\geqslant0}$ be the coefficien…

Let $A\subset\mathbb{Z}_{>0}$, let $p^{(k)}(n)$ be the generalized partition functions associated with $A$, and define … Assume that $A$ has properties $P_{k+1},P_{k+2},\dots$. Eve…

Higher-order modified Alder conjecture. For all $d,n\geq 1$,

Kang and Park's conjecture. For all $d,n\geq 1$,

Conjecture centrale. Pour tout entier $n\geq1$,

Nonnegative product conjecture.

Let $M(n)$ be the number of distinct multinomial coefficients arising from partitions of $n$, and let $p^*(n)$ be the total number of partitions of $n$ into parts that are either a…

First Borwein conjecture. The polynomials $A_n(q)$, $B_n(q)$ and $C_n(q)$ have nonnegative coefficients.

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…

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…

Let $T_o(q)=\sum_{n\geq 0}t_o(n)q^n$, and let $n$ be a nonnegative integer. Parity conjecture. If $8n+9$ has a prime divisor $p\equiv 5,7\pmod{8}$ to odd exponent, equivalently if…

Let $C(q)=\sum_{n\geq 0}c(n)q^n$ be the companion series whose coefficients are denoted by $c(n)$. Unboundedness conjecture. … The conjecture asserts that the coefficients of…

Andrews–Merca–Guo–Zeng conjecture. This coefficient is nonnegative.

Let $b_2”(n)$ be the signed count associated with partitions whose parts greater than $1$ lie in $[2k,2l-1]$, whose smallest even part is $2k$, whose number of ones is $l-1$, and w…