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On the number of partitions of the hypercube ${\bf Z}_q^n$ into large subcubes

Yuriy Tarannikov

TL;DR

The paper addresses counting partitions of the $q$-ary hypercube ${\bf Z}_q^n$ into $q^m$ subcubes of dimension $n-m$ as $n\to\infty$. It introduces star matrices and the bang operation, along with fractal star matrices, to analyze expandability and isolate the dominant structures. It proves exact values $N^{\rm coord}_q(m)=\frac{q^m-1}{q-1}$ and $c^{\rm coord*}_q\left(\frac{q^m-1}{q-1},m\right)=\left(\frac{q^m-1}{q-1}\right)!$, and derives the main asymptotic $c^{\rm coord}_q(n,m)\sim n^{(q^m-1)/(q-1)}$; this progress connects to ABDs and provides a precise enumeration in the large-$n$ regime. The framework shows that fractal constructions govern growth and offers a structural path to counting large-dimension cube partitions in ${\bf Z}_q^n$.

Abstract

We prove that the number of partitions of the hypercube ${\bf Z}_q^n$ into $q^m$ subcubes of dimension $n-m$ each for fixed $q$, $m$ and growing $n$ is asymptotically equal to $n^{(q^m-1)/(q-1)}$. For the proof, we introduce the operation of the bang of a star matrix and demonstrate that any star matrix, except for a fractal, is expandable under some bang, whereas a fractal remains to be a fractal under any bang.

On the number of partitions of the hypercube ${\bf Z}_q^n$ into large subcubes

TL;DR

The paper addresses counting partitions of the -ary hypercube into subcubes of dimension as . It introduces star matrices and the bang operation, along with fractal star matrices, to analyze expandability and isolate the dominant structures. It proves exact values and , and derives the main asymptotic ; this progress connects to ABDs and provides a precise enumeration in the large- regime. The framework shows that fractal constructions govern growth and offers a structural path to counting large-dimension cube partitions in .

Abstract

We prove that the number of partitions of the hypercube into subcubes of dimension each for fixed , and growing is asymptotically equal to . For the proof, we introduce the operation of the bang of a star matrix and demonstrate that any star matrix, except for a fractal, is expandable under some bang, whereas a fractal remains to be a fractal under any bang.

Paper Structure

This paper contains 8 sections, 13 theorems, 16 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Star matrices of partitions and their properties
  3. An obvious lower asymptotic bound for the number of partitions
  4. Fractal star matrices and their properties
  5. Operation of the bang of a star matrix
  6. Non-expandable star matrices
  7. Main Theorems
  8. Conclusion

Key Result

Theorem 1

Tar22 Let $q\ge 2$. For any natural $m$ there exists the smallest natural $N$ such that for $n>N$ there are no A-primitive partitions of ${\bf Z}_q^n$ into $q^m$ subcubes of dimension $n-m$.

Figures (7)

  • Figure 1: Star matrix with stars in all columns.
  • Figure 2: Recursive construction of a fractal matrix.
  • Figure 3: Examples of fractal matrices.
  • Figure 4: An example of a transfractal is highlighted with a black frame.
  • Figure 5: An example of a bang of a star matrix.
  • ...and 2 more figures

Theorems & Definitions (30)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 6
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 20 more