39 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…

Let $s$ and $t$ be positive integers with $\gcd(s,t)=1$, and let $f_t(s)$ be the polynomial appearing in Fayers' formula for the number of $(s,s+t,s+2t,\dots)$-core partitions. Fay…

Let $N\in\mathbb{N}$ and $y\in\Gamma_N$. Let $\eta\in X_{N,y}(\mathbb{Z})$ be an integral point. For $\widehat{w}\in\Sigma W^0$ and $g\in G$, define the action $g\cdot\widehat{w}$…

For positive integers $s$ and $k$, a partition is a self-conjugate $(s,s+1,\ldots,s+k)$-core partition if it is equal to its conjugate and is simultaneously an $s$-core, an $(s+1)$…

Let $s$ and $k$ be positive integers. A self-conjugate $(s,s+1,\dots,s+k)$-core is a self-conjugate partition that is simultaneously an $s$-core, $(s+1)$-core, and so on through an…

Sahin's extended conjecture. For any positive integers $r\leq d$, we have

For integers $s$, $r$, and $d$ with $1\le r\le d$, let $N_{d,r}(s)$ denote the number of $(s,s+r)$-core partitions with $d$-distinct parts. Recurrence conjecture. The values of…

Let $s$ be a positive integer and let $a,b$ satisfy $0\leq a,b<s$. For parameters $r$ and $c$, write $\mathcal{C}_{(r\,|\,0,c)}$ for the corresponding set of bipartitions, and let…

Joint pre-moment conjecture. The $k$th pre-moment $m_k(d,n)$ of $X_{d,n}$ is of the form

Fixed-$d$ normality conjecture. For fixed $d$, the distribution of $X_{d,n}$ is asymptotically normal. That is,

Generalized-Fibonacci pre-moment conjecture. For each $d$, the $k$th pre-moment $m_k(d,n)$ of $X_{d,n}$ is of the form

Two-scale normality conjecture. For each fixed $n$, the distribution of $X_{d,n}$ is not asymptotically normal; in fact, $(X_{d,n}-\mu_{d,n})/\sigma_{d,n}$ tends to some abnormal d…

Polynomial pre-moment conjecture. For each $n$, the $k$th pre-moment $m_k(d,n)$ of $X_{d,n}$ is a polynomial in $d$. Further, the degree of this polynomial is…

Let $s$ and $m$ be positive integers, and let $t$ be determined by either $s=2t-1$ or $s=2t-2$. A maximal $(s,ms-1,ms+1)$-core partition is a core partition maximal under inclusion…

For each positive integer $s$, let $X_s$ be the size of a uniformly chosen $(s,s+1)$-core partition with distinct parts. Let $\mathbb{E}[X_s]$ and $\operatorname{Var}(X_s)$ denote…

Let $s$ be an odd positive integer. An $(s,s+2)$-core partition is a partition that is simultaneously an $s$-core and an $(s+2)$-core, and its parts are required to be distinct. St…

Let $s$ be odd, and consider $(s,s+2)$-core partitions into distinct parts. Power-of-two conjecture. Their number equals … The paper presents this as a proposed extension beyond th…

Let an $(s,t)$-core partition be a partition that is simultaneously $s$-core and $t$-core, and let $F_n$ denote the $n$th Fibonacci number. Amdeberhan's conjecture. The number of…

Let $\widetilde{W}$ be a simply-laced affine Weyl group with Coxeter number $h$, let $b$ be coprime to $h$, and let $\mathcal{S}_\Phi(b)$ be the relevant Sommers region. Let…

Let $p$ and $q$ be distinct odd primes. A partition is a $(\bar p,\bar q)$-core partition if it is simultaneously a $\bar p$-core and a $\bar q$-core, and it is even when its corre…

Armstrong's rank and co-skew-length symmetry conjecture. The following $q,t$-polynomials are equal:

Let $m$ and $n$ be coprime positive integers. An $(m,n)$-core is a partition that is simultaneously an $m$-core and an $n$-core; denote the set of such cores by $\mathcal P_{m,n}$.…

Weighted self-conjugate core-average conjecture.

Coset age conjecture. There is an age function $\iota$ on the cosets $\mathfrak{c}\in\boldsymbol{\Lambda}/\boldsymbol{\Lambda}_T$ such that

Let $a$ and $b$ be positive integers, and let an $(a,b)$-core partition be a partition with no hook length divisible by either $a$ or $b$. For a partition $\lambda$, write…