Enumeration of pattern-avoiding $(0,1)$-matrices and their symmetry classes

TL;DR

This paper explores the structure and enumeration of pattern-avoiding (0,1)-matrices, establishing bijections with plane partitions, classifying symmetry classes, and deriving formulas for skew shape fillings.

Contribution

It introduces a new lattice path perspective, classifies symmetry classes, and provides explicit enumeration formulas for pattern-avoiding matrices and their skew shape fillings.

Findings

Maximal I_k-avoiding matrices are equinumerous with certain plane partitions.

Enumeration formulas for symmetry classes are given by simple product formulas.

A conceptual formula for skew shape fillings is derived.

Abstract

Recently, Brualdi and Cao studied $I_k$-avoiding $(0,1)$-matrices by decomposing them into zigzag paths and proved that the maximum number of $1$'s in such a matrix is given by an exact formula. We further study the structure of maximal $I_k$-avoiding $(0,1)$-matrices (IAMs) by interpreting them as families of non-intersecting lattice paths on the square lattice. Using this perspective, we establish a bijection showing that IAMs are equinumerous with plane partitions of a certain size. Moreover, we classify all ten symmetry classes of IAMs under the action of the dihedral group of order $8$ and show that the enumeration formulas for these classes are given by simple product formulas. Extending this approach to skew shapes, we derive a conceptual formula for enumerating maximal $I_k$-avoiding $(0,1)$-fillings of skew shapes.