Some inequalities for the multicolor restricted partition function $p_\mathcal{A}(n,k)$

Abstract: For a non-decreasing sequence of positive integers $\mathcal{A}=\left(a_i\right)_{i=1}^\infty$ and a fixed integer $k\geqslant1$, the multicolor restricted partition function $p_\mathcal{A}(n,k)$ counts the number of partitions of $n$ with parts in the multiset $\{a_1,a_2,\ldots,a_k\}$. The talk is devoted to some multiplicative inequalities related to $p_\mathcal{A}(n,k)$. Among other things, we will examine: the Bessenrodt-Ono inequality for $p_\mathcal{A}(n,k)$, the $\log$-concavity of the sequence $\left(p_\mathcal{A}(n,k)\right)_{n=1}^\infty$, the higher order Tur\'an property and other similar phenomena.

number theory

Audience: researchers in the discipline

Combinatorial and additive number theory (CANT 2022)