317 problems

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…

Let $\overline{p}(n)$ denote the number of overpartitions of $n$. For $n,k\geq 1$, Guo–Zeng's conjecture. … with strict inequality if $n\geq k^2$. The conjecture was proved indepen…

For integers $d\geq 1$, consider the partial fraction decomposition … Here $C_{h,k,\ell}(d)$ denotes the coefficient associated with the reduced root of unity $e^{2\pi i h/k}$ and…

A Beatty sequence is a sequence of the form $\{\lfloor n\alpha\rfloor:n\in\mathbb{N}\}$ with modulus $\alpha>1$. A partition of $\mathbb{N}$ into $m$ Beatty sequences has distinct…

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…

Garvan's conjecture. For all $k>1$ and $n\geq 1$,

Andrews–Bachraoui conjecture. There holds

Let $\mathcal O(n)$ be the set of odd partitions of $n$, meaning partitions all of whose parts are odd, and let $\mathrm{num}_{\mathcal O}(n,x)$ be the associated numerator polynom…

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 $q(n)$ denote the number of partitions of $n$ into distinct parts. A sequence is log-concave when $q(n)^2\geq q(n-1)q(n+1)$, and the third-order Turán inequalities are the corr…

Let $p_k(n)$ denote the number of $k$-coloured partitions of $n$. Chern–Fu–Tang's conjecture. For $1\leq\ell<n$ and $k\geq 2$, except for $(k,n,\ell)=(2,6,4)$, one has … This conje…

Let $\mathfrak g$ be the Kac–Moody Lie algebra under consideration and let $d$ be the imaginary root in the example, with $d=(n,n,1)$. Write $\mathfrak g_d$ for the root space of d…

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…

Let … Here $q$ is a complex number with $|q|<1$, so the double Lambert series defines an analytic function in its domain of convergence. Andrews–Dixit–Schultz–Yee conjecture. The f…

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…

For integers $t\geq 2$ and $k\geq 1$, let $b_{t,k}(n)$ denote the number of hooks of length $k$ in all the $t$-regular partitions of $n$. Singh–Barman's hook-length bias conjecture…

Let $a_t(n)$ be the total number of $t$-hooks in all partitions of $n$ into odd parts, and let $b_t(n)$ be the total number of $t$-hooks in all partitions of $n$ into distinct part…

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 … where $\lambda$ runs over Young tableaux of size $n$ and $H(\lambda)$ is the set of hook lengths associated with $\lambda$. Heim–Neuhauser conjecture. The polynomials…

Let $0<s<1$, let $\mathcal{C}_1,\mathcal{C}_2,\ldots$ be independent events with probabilities $\mathbf{P}_s(\mathcal{C}_n)=1-e^{-ns}$, and let $A_k$ be the event that there is no…

A partition is t-core if it has no hook length divisible by $t$. The $t$-core conjecture. For every integer $j\geq 0$ and every integer $t\geq 4$, there exists a $t$-core partition…

Let $b_{t,i}(n)$ denote the number of hooks of length $i$ in all $t$-regular partitions of $n$. Singh–Barman's conjecture. For $t\ge 3$ and all $n\ge 0$,…