4 problems

Let $r_{{\cal{MN}}_n}(t)$ be the rank-generating polynomial of the set ${\cal{MN}}_n$ of monotonic triangles, and let $\operatorname{ASM}(n)$ denote the number of alternating sign…

Let $F_{\lambda}(q)=\sum_{i=0}^{d}c_iq^i$ be the rank-generating function for partitions contained inside a partition $\lambda$. The polynomial is flawless if $c_i\le c_{d-i}$ for…

Let $F_{\lambda}(q)$ be the rank-generating function for partitions contained inside a partition $\lambda$. A polynomial is unimodal if its coefficients weakly increase and then we…

Let $F_{\lambda}(q)$ be the rank-generating function for partitions contained inside a partition $\lambda$, and let … be a partition whose parts form an arithmetic progression. A p…