57 problems

Let $s$ be an integer with $s\geq 1$, and consider $(s,s+1,s+2)$-core partitions. The Amdeberhan conjecture. If $s=2t-1$, the largest size is … if $s=2t-2$, the largest size is … T…

Let $s_1$ and $s_2$ be coprime positive integers, and consider the finite set of $(s_1,s_2)$-core partitions. The Armstrong conjecture. The average size of these partitions equals…

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…

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…

A partition is an $s$-core partition if none of its hook lengths is divisible by $s$; an $(s_1,s_2,\cdots,s_m)$-core partition is simultaneously an $s_i$-core partition for every…

Let $s$ be a positive integer, and consider $(s,s+2)$-core partitions with distinct parts. The Straub conjecture. The largest size of such a partition equals … The conjecture was f…

Let $a$ and $b$ be coprime positive integers. An $(a,b)$-core partition is a partition that is simultaneously an $a$-core and a $b$-core, meaning that no cell in its Ferrers diagra…

Stanton's monotonicity conjecture.

Let $\boldsymbol{s}=(0,1,2,3,4)$ and let $Q_{\boldsymbol{s}}$ and $\mathcal{O}_{\boldsymbol{s}}$ be as above. Exceptional-value conjecture. For $n=5$, the only nonnegative integer…

Refined universality conjecture. Assume $n\geq6$. Then $Q_{\boldsymbol{s}}(t_1,\ldots,t_n)$ is universal on $\mathcal{O}_{\boldsymbol{s}}$. Computational evidence supports the clai…

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 $s$ and $t$ be positive integers with $\gcd(s,t)=1$. An $(s,s+t,s+2t,\dots)$-core partition is a partition whose hook lengths avoid every integer in the progression…

Existence question for boundary t-cores. When is $\widetilde{P}^t_{r,s}>0$?

Let $t$ be a positive integer. Write $\operatorname{c}_t(n)$ for the number of $t$-core partitions of $n$, and $\operatorname{sc}_{2t-1}(n)$ for the number of self-conjugate…

Homogeneity conjecture. If $\lambda$ is a $2$-core, then $\operatorname{spn}\lambda$ is homogeneous.

Let $s$ and $t$ be coprime positive integers, and let $\lambda$ range over the $(s,t)$-core partitions. Write $n(\lambda)$ for the number of parts of $\lambda$ and $m(\lambda)$ for…

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…

Moment rationality conjecture. For fixed $n$, $M_k(d,n)$ is a rational function in $d$.

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