An integration of Euler's pentagonal partition

TL;DR

This paper introduces a new recurrence relation for counting compositions of positive integers, inspired by Euler's pentagonal number theorem, using a novel discrete integration approach.

Contribution

It presents a new recurrence formula for compositions, connecting Euler's pentagonal numbers with a discrete integration method, supported by bijective and generating function proofs.

Findings

New recurrence relation for compositions

Equivalence proven via bijective and generating function methods

Connects Euler's pentagonal numbers with discrete integration

Abstract

A recurrent formula is presented, for the enumeration of the compositions of positive integers as sums over multisets of positive integers, that closely resembles Euler's recurrence based on the pentagonal numbers, but where the coefficients result from a discrete integration of Euler's coefficients. Both a bijective proof and one based on generating functions show the equivalence of the subject recurrences.