Even and odd compositions with restricted parts

TL;DR

This paper extends Legendre's and Euler's classical results on partitions to compositions, providing new insights and generalizations related to integer sequences in the OEIS.

Contribution

It introduces an analogous result for compositions with restricted parts and explores generalizations connected to OEIS entries.

Findings

Difference between counts of even and odd compositions is 0, 1, or -1

Generalizations related to various OEIS sequences

Extension of classical partition results to compositions

Abstract

A result of Legendre asserts that the difference between the numbers of (length) even and odd partitions of $n$ into distinct parts is $0$, $1$, or $-1$; this also follows from Euler's pentagonal number theorem. We establish an analogous result for compositions and obtain some generalizations that are related to various entries in the On-Line Encyclopedia of Integer Sequences.