# Enumeration and Asymptotic Formulas for Rectangular Partitions of the Hypercube

## TL;DR

This paper introduces a generalized enumeration of hypercube subdivisions into rectangular blocks, providing explicit formulas, asymptotic analysis, and a combinatorial proof linking these counts to full p-ary trees, extending Catalan number concepts.

## Contribution

It offers explicit formulas and asymptotic behavior for the generalized hypercube partition numbers, and provides a new elementary combinatorial proof of their functional equation.

## Key findings

- Derived simple finite sum formulas for $C_{d,p}(n)$
- Determined growth rate and asymptotic behavior of $C_{d,p}(n)$
- Established a bijection between hypercube decompositions and full p-ary trees

## Abstract

We study a two-parameter generalization of the Catalan numbers: $C_{d,p}(n)$ is the number of ways to subdivide the $d$-dimensional hypercube into $n$ rectangular blocks using orthogonal partitions of fixed arity $p$. Bremner \& Dotsenko introduced $C_{d,p}(n)$ in their work on Boardman--Vogt tensor products of operads; they used homological algebra to prove a recursive formula and a functional equation. We express $C_{d,p}(n)$ as simple finite sums, and determine their growth rate and asymptotic behaviour. We give an elementary proof of the functional equation, using a bijection between hypercube decompositions and a family of full $p$-ary trees. Our results generalize the well-known correspondence between Catalan numbers and full binary trees.