3 problems

Let $v(n)$ count odd-balanced unimodal sequences of size $2n+2$, and let $v(m,n)$ count those with rank $m$. Let $\ell\not\equiv-1\pmod8$ be prime and let $k$ be a positive integer…

Bryson–Ono–Pitman–Rhoades conjecture. For all $n$,

The unimodal rank $u(m,n)$ counts unimodal sequences of weight $n$ and rank $m$; three further unimodal ranks are denoted by $w(m,n)$, $v(m,n)$, and $\nu(m,n)$. For non-negative in…