5 problems

Let $M(a,c;n)$ denote the number of partitions of $n$ whose crank is congruent to $a$ modulo $c$. Andrews–Lewis's conjecture. For every $n\in\mathbb N$, and for…

Let $N(k,11,n)$ and $M(k,11,n)$ denote the rank and crank counting functions modulo $11$. For integers $a_k,b_k$ with $0\leq k\leq5$, suppose … For a residue $m$ modulo $11$, defin…

Let $N(r,11,n)$ and $M(r,11,n)$ denote the numbers of partitions of $n$ with rank and crank, respectively, congruent to $r$ modulo $11$. For each residue $m$ modulo $11$, write…

Let $p(n)$ be the number of partitions of $n$, and let $M(r,11,n)$ denote the number of partitions of $n$ whose crank is congruent to $r$ modulo $11$. Ekin's crank inequality. For…

For an integer $k\geq 1$, let $M_{2k}(n)$ and $N_{2k}(n)$ be the $2k$-th crank and rank moments, respectively, let $p(n)$ be the partition function, and define … The constants sati…