The (Symbolic and Numeric) Computational Challenges of Counting 0-1 Balanced Matrices

TL;DR

This work investigates counting balanced $0$-$1$ matrices with a fixed numeric dimension in rows and a symbolic number of columns, formalizing three core problems: the vanilla balanced case, and restricted versions with forbidden horizontal and vertical patterns. It proves that the sequence $b_k(n)$, counting $2k\times 2n$ balanced matrices, is holonomic (a linear recurrence with polynomial coefficients) and extends this holonomic property to general pattern-avoidance counts $b_{H,V,k}(n)$; in the unrestricted setting with a finite alphabet, the generating functions are rational (C-finite). The authors couple symbolic methods (holonomic systems, transfer-matrix, Goulden-Jackson) with heavy numeric computations to produce recurrences for small $k$ (e.g., $k=2$ and $k=3$), conjecture recurrences for $k=4$, and push term-generation far beyond humans’ reach using public Maple/C and KK22-based approaches. They illustrate the practical side with Not-Alone puzzles, showing exact counts for small puzzles and highlighting significant computational challenges for larger ones, while releasing software and data to support further exploration. Overall, the paper maps the landscape between provable holonomic structure and the intense numeric effort required to pin down high-order recurrences, underscoring the OEIS as a central repository for these hard-to-compute combinatorial sequences.

Abstract

A chessboard has the property that every row and every column has as many white squares as black squares. In this mostly methodological note, we address the problem of counting such rectangular arrays with a fixed (numeric) number of rows, but an arbitrary (symbolic) number of columns. We first address the ``vanilla" problem where there are no restrictions, and then go on to discuss the still-more-challenging problem of counting such binary arrays that are not permitted to contain a specified (finite) set of horizontal patterns, and a specified set of vertical patterns. While we can rigorously prove that each such sequence satisfies some linear recurrence equation with polynomial coefficients, actually finding these recurrences poses major {\it symbolic}-computational challenges, that we can only meet in some small cases. In fact, just generating as many as possible terms of these sequences is a big {\it numeric}-computational challenge. This was tackled by computer whiz Ron H. Hardin, who contributed several such sequences, and computed quite a few terms of each. We extend Hardin's sequences quite considerably. We also talk about the much easier problem of counting such restricted arrays without balance conditions.

Theorems & Definitions (10)

• Theorem 1
• proof
• Definition 1
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• Theorem 5
• Conjecture 6