7 problems

Hou–Jagadeesan's maximal-value conjecture. The following are true:

Let $N(r,2;n)$ denote the number of partitions of $n$ whose rank is congruent to $r$ modulo $2$, for $r\in\{0,1\}$. Hou–Jagadeesan's conjecture. If $r=0$ (respectively, $r=1$), the…

Let $N(m,n)$ and $M(m,n)$ denote, respectively, the rank and crank counting functions for ordinary partitions of $n$ with statistic $m$. Strict log-concavity conjecture. The follow…

Let $u(m,n)$ denote the number of strongly unimodal sequences of size $n$ and rank $m$. Strict log-concavity conjecture. For $n\ge37$ and $|m|\le n-23$, we have … The conjecture is…

Wei–Zhang inequalities. For $n\geq 11$,

Ji–Zhang–Zhao inequalities. For $n\geq 0$ and $1\leq i\leq 4$,

Let $\overline{N}(m,n)$ denote the number of overpartitions of $n$ with $D$-rank $m$, and let $\overline{N2}(m,n)$ denote the number of overpartitions of $n$ with $M_2$-rank $m$. C…