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Enumeration and Asymptotic Formulas for Rectangular Partitions of the Hypercube

Yu Hin Au, Fatemeh Bagherzadeh, Murray R. Bremner

TL;DR

This work extends the Catalan paradigm by introducing $C_{d,p}(n)$, counting orthogonal $p$-ary partitions of the unit $d$-cube and linking to higher-arity interchange laws and algebraic operads. It delivers a concise finite-sum formula via Lagrange inversion, derives detailed asymptotics with a universal polynomial $q$ governing growth, and provides a purely combinatorial proof of the BD functional equation. A central contribution is the combinatorial interpretation of $C_{d,p}(n)$ in terms of interchange maximal full $p$-ary trees, establishing a deep connection between hypercube decompositions and tree structures. The paper also sketches future directions in Wedderburn–Etherington numbers and operad theory, highlighting rich structural connections and open problems.

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.

Enumeration and Asymptotic Formulas for Rectangular Partitions of the Hypercube

TL;DR

This work extends the Catalan paradigm by introducing , counting orthogonal -ary partitions of the unit -cube and linking to higher-arity interchange laws and algebraic operads. It delivers a concise finite-sum formula via Lagrange inversion, derives detailed asymptotics with a universal polynomial governing growth, and provides a purely combinatorial proof of the BD functional equation. A central contribution is the combinatorial interpretation of in terms of interchange maximal full -ary trees, establishing a deep connection between hypercube decompositions and tree structures. The paper also sketches future directions in Wedderburn–Etherington numbers and operad theory, highlighting rich structural connections and open problems.

Abstract

We study a two-parameter generalization of the Catalan numbers: is the number of ways to subdivide the -dimensional hypercube into rectangular blocks using orthogonal partitions of fixed arity . Bremner \& Dotsenko introduced 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 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 -ary trees. Our results generalize the well-known correspondence between Catalan numbers and full binary trees.

Paper Structure

This paper contains 16 sections, 12 theorems, 71 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Catalan numbers
  3. Geometry of higher-dimensional Catalan numbers
  4. Interchange laws for operations of higher arity
  5. Outline of this paper
  6. Enumeration formulas
  7. Asymptotic behavior and growth rate
  8. Asymptotic behavior
  9. Growth rate
  10. Interpretation of $C_{d,p}(n)$ in terms of $p$-ary trees
  11. A combinatorial proof of Theorem \ref{['BDTheorem']}
  12. Interchange maximal trees
  13. Future research directions
  14. Wedderburn--Etherington numbers
  15. Algebraic operads
  16. ...and 1 more sections

Key Result

Theorem 3

Define the generating function Then $y$ satisfies this polynomial functional equation:

Figures (6)

  • Figure 1: Decompositions of the unit square ($d=2)$ using bisections ($p=2$) into $n \le 4$ subrectangles
  • Figure 2: Graphs of the polynomials $q_{d,p}(z)$ for $(d,p) = (2,2), (3,2), (2,3), (3,3)$
  • Figure 3: The growth rate $\mathcal{G}_{d,p}$ for various $d$ and $p$
  • Figure 4: Illustrating the definition of $D_S$ in the case of $d=p=2$
  • Figure 5: Example of the map $f$ from trees to decompositions (Definition \ref{['treetodecomp']})
  • ...and 1 more figures

Theorems & Definitions (30)

  • Definition 1
  • Definition 2
  • Theorem 3: BD, §3.1
  • Proposition 4
  • proof
  • Corollary 5
  • proof
  • Corollary 6
  • proof
  • Definition 7
  • ...and 20 more