Odd unimodal sequeneces

TL;DR

This paper investigates odd unimodal and strongly unimodal sequences using q-series and asymptotic analysis, deriving generating functions, asymptotic formulas, and congruences, thereby advancing understanding of their combinatorial properties.

Contribution

It introduces new generating functions, asymptotic formulas, and congruences for odd unimodal sequences, expanding the theoretical framework in this area.

Findings

Derived fundamental generating functions for odd unimodal sequences.

Obtained asymptotic main terms using Euler--Maclaurin summation.

Discovered families of congruences modulo 4 for odd strongly unimodal sequences.

Abstract

In this paper we study odd unimodal and odd strongly unimodal sequences. We use $q$-series methods to find several fundamental generating functions. Employing the Euler--Maclaurin summation formula we obtain the asymptotic main term for both types of sequences. We also find families of congruences modulo $4$ for the number of odd strongly unimodal sequences.