On the number of hypercubic bipartitions of an integer

TL;DR

This paper analyzes a classic divide-and-conquer recurrence related to hypercubic bipartitions, providing a new characterization, recursive formulas, and explicit solutions for counting maximum-yielding bipartitions.

Contribution

It introduces a novel characterization of bipartitions that maximize the recurrence and derives formulas and generating functions for counting them.

Findings

Derived recursive formulas for h(n)

Provided a generating function h(x)

Obtained an explicit formula for h(n)

Abstract

We revisit a well-known divide-and-conquer maximin recurrence $f(n) = \max(\min(n_1,n_2) + f(n_1) + f(n_2))$ where the maximum is taken over all proper bipartitions $n = n_1+n_2$, and we present a new characterization of the pairs $(n_1,n_2)$ summing to $n$ that yield the maximum $f(n) = \min(n_1,n_2) + f(n_1) + f(n_2)$. This new characterization allows us, for a given $n\in\nats$, to determine the number $h(n)$ of these bipartitions that yield the said maximum $f(n)$. We present recursive formulae for $h(n)$, a generating function $h(x)$, and an explicit formula for $h(n)$ in terms of a special representation of $n$.