Distinct parts partitions without sequences

TL;DR

This paper investigates partitions into distinct parts without consecutive integers, providing new generating function representations, asymptotic formulas, and combinatorial inequalities, thus advancing understanding of their combinatorial and analytical properties.

Contribution

It introduces a double series representation for the generating function and derives asymptotic formulas and inequalities for these special partitions, expanding prior combinatorial studies.

Findings

Double series representation for the generating function

Asymptotic formula for enumeration function

Several combinatorial inequalities

Abstract

Partitions without sequences of consecutive integers as parts have been studied recently by many authors, including Andrews, Holroyd, Liggett, and Romik, among others. Their results include a description of combinatorial properties, hypergeometric representations for the generating functions, and asymptotic formulas for the enumeration functions. We complete a similar investigation of partitions into distinct parts without sequences, which are of particular interest due to their relationship with the Rogers-Ramanujan identities. Our main results include a double series representation for the generating function, an asymptotic formula for the enumeration function, and several combinatorial inequalities.