23 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…

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…

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 $n\ge 0$,…

Han's conjecture. For every $k\in\mathbb{N}$, $P(n)$ is a polynomial in $n$.

Let $\lambda$ range over partitions of $n$. For a box $\square\in\lambda$, let $h_{\square}$ denote its hook length, and let $f_\lambda$ denote the number of standard Young tableau…

Let $r_{1,t}(n)$ and $r_{2,t}(n)$ denote the corresponding total numbers of $t$-hooks for the two partition classes arising from the first Rogers–Ramanujan identity, and let…

Let $b_{t,k}(n)$ denote the number of hooks of length $k$ across all $t$-regular partitions of $n$. Singh and Barman's conjecture. For every integer $t\geq 3$ and every $n\geq 0$,…

Let $b_{t,i}(n)$ denote the number of hooks of length $i$ in all $t$-regular partitions of $n$. Even-parameter hook bias conjecture. For even $k\ge 8$, $b_{2,k}(n)\ge b_{2,k+1}(n)$…

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

Ballantine et al.'s conjecture. For every $k\geq2$, there exists an integer $N_k$ such that

Ballantine--Burson--Craig--Folsom--Wen hook length bias conjecture. For every integer $t\geq 2$, both of the following hold:

Beck's hook length bias conjecture. For all $n\geq 0$,

Let $a_t^{\star}(n)$ denote the number of hooks of length $t$ among all self-conjugate partitions of $n$. For integers $n\geq0$ and $m\geq1$, define divisibility in the usual sense…

Let $a_h^*(n)$ denote the number of hooks of length $h$ among self-conjugate partitions of $n$. The even-hook divisibility conjecture. For every integer $n\geq0$ and every integer…

Let $a_h^*(n)$ and $b_h^*(n)$ denote the numbers of hooks of length $h$ among self-conjugate partitions of $n$ and partitions of $n$ into distinct odd parts, respectively. The hook…

The hook-number ratio conjecture. For fixed $h\geq 2$, there exists an integer $N_h>0$ such that $a_h(n)\geq b_h(n)$ for all $n>N_h$, and there exists a constant $\gamma_h>1$ such…

Even-hook divisibility conjecture. For all integers $n\geq 0$ and $m\geq 1$,

Odd-versus-distinct hook bias conjecture. For every integer $t\geq 2$, there exists an integer $N_t$ such that for all $n>N_t$, $a_t(n)\geq b_t(n)$. The conjectured values are

Let $S'(n)$ count the number of partitions of size at most $n$ that are in the image of Keith and Nath's map from numerical semigroups to integer partitions, and let $P(n)$ count a…

Let $\sigma\subset\tau$, where $\tau=(\ell^k)$ is a rectangle, and let $\sigma^\ast$ be the $180$-degree rotation of $\sigma$. Form the skew shape $\lambda/\mu$ by attaching two co…

Amdeberhan's conjecture. The generating function for the sequence $b_n$ is

Conjecture 5.2. The generating function satisfies