On flushed partitions and concave compositions

TL;DR

This paper provides combinatorial proofs for generating functions related to flushed partitions and concave compositions, interprets a problem posed by Sylvester, and proves several identities including those of Ramanujan and mock theta functions.

Contribution

It introduces new combinatorial involutions and interpretations for complex partition identities and mock theta functions, expanding understanding of these mathematical objects.

Findings

Proved generating functions for flushed partitions and concave compositions.

Provided combinatorial interpretation and proof of Sylvester's problem.

Established new identities including Ramanujan's and mock theta functions.

Abstract

In this work, we give combinatorial proofs for generating functions of two problems, i.e., flushed partitions and concave compositions of even length. We also give combinatorial interpretation of one problem posed by Sylvester involving flushed partitions and then prove it. For these purposes, we first describe an involution and use it to prove core identities. Using this involution with modifications, we prove several problems of different nature, including Andrews' partition identities involving initial repetitions and partition theoretical interpretations of three mock theta functions of third order $f(q)$, $\phi(q)$ and $\psi(q)$. An identity of Ramanujan is proved combinatorially. Several new identities are also established.