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

TL;DR

The paper establishes a deep link between maximal $I_k$-avoiding $(0,1)$-matrices and plane partitions via non-intersecting lattice paths, proving $|\\mathcal{M}_{m,n;k}|=H(m-k+1,n-k+1,k-1)$ where $H$ is MacMahon’s product. It then classifies all $D_8$ symmetry classes of IAMs, deriving simple product formulas for five classes tied to plane-partition symmetry and showing parity-dependent trivial counts for the remaining five. Extending these ideas, it provides a determinant-based framework for counting maximal $I_k$-avoiding fillings of skew shapes, with a concrete product formula in a staircase-removed rectangle case and a general determinant expression for arbitrary skew shapes. The results illuminate structural parallels between IAMs, plane partitions, and skew-shape fillings, offering tools for weighted enumeration and potential bijections across symmetry classes, while suggesting avenues for further pattern-avoidance extensions. Overall, the work combines combinatorial bijections, non-intersecting paths, and determinant evaluations to yield elegant product formulas and determinant expressions for a broad class of pattern-avoiding objects.

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 number. 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.

Figures (5)

• Figure 1: (a) A plane partition $\pi \in \mathcal{PP}(5,3,4)$. (b) Viewing $\pi$ as a pile of unit cubes in the $5 \times 3 \times 4$ box. (c) The non-intersecting paths encoding of $\pi$. (d) The corresponding non-intersecting lattice paths of $\pi$ which is obtained from deforming the paths in Figure \ref{['fig.pppath']}.
• Figure 2: An illustration of two staircases of size $k-1$ in a matrix. The "diagonal" entries of staircases are labeled $\ell_i$'s and $r_i$'s, and the entries marked $*$ must be filled with $1$'s.
• Figure 3: Two different decompositions of a matrix in $\mathcal{M}_{9,7;5}$ into $4$ zigzag paths.
• Figure 4: An example of extending zigzag paths in a decomposition of $\mathcal{Z}_6$ (left) to a decomposition of $\mathcal{Z}_7$ (right).
• Figure 6: (a) A maximal $I_5$-avoiding $(0,1)$-filling of $\bar{R}_{9,12;t}$, where $t=9-5$. (b) The corresponding non-intersecting lattice paths. Note that paths can only stay weakly above the blue line.

Theorems & Definitions (8)

• proof
• proof
• proof : Proof of Theorem \ref{['thm1']}.
• proof
• proof : Proof of Corollary \ref{['cor.product']}
• proof : Proof of Theorem \ref{['thm2']}.
• proof : Proof of Theorem \ref{['thm3']}
• proof : Proof of Theorem \ref{['thm4']}