Pattern Forcing (0,1)-Matrices
Lei Cao, Shen-Fu Tsai
TL;DR
The paper studies how a fixed $(0,1)$-matrix pattern $Q$ can be forced or strongly forced inside larger $(0,1)$-matrices. It defines two extremal functions, $\mathrm{m}(m,n,Q)$ and $\mathrm{M}(m,n,Q)$, corresponding to the minimum possible number of $1$-entries in a $Q$-forcing matrix and the maximum possible number of $1$-entries in a strongly $Q$-forcing matrix. For $Q$-forcing, it proves the existence and uniqueness of the extremal construction, expresses $\mathrm{m}(m,n,Q)$ via corner functions and the core $Q_{core}$, and connects these to Young diagrams to obtain exact formulas and monotonicity. For strongly $Q$-forcing, it shows $mn-\mathrm{M}(m,n,Q)=O(m+n)$ and determines $\mathrm{M}(n,Q)$ for all $2\times 2$ and $3\times 3$ permutation matrices, identifying dihedral symmetry classes and presenting a conjectural formula for $\mathrm{M}(n,I_k)$ with special cases $k=2,3$ verified. Overall, the work highlights a sharp contrast between forcing and strong forcing and advances the combinatorial theory of pattern embedding in $(0,1)$-matrices.
Abstract
We introduce two related notions of pattern enforcement in $(0,1)$-matrices: $Q$-forcing and strongly $Q$-forcing, which formalize distinct ways a fixed pattern $Q$ must appear within a larger matrix. A matrix is $Q$-forcing if every submatrix can realize $Q$ after turning any number of $1$-entries into $0$-entries, and strongly $Q$-forcing if every $1$-entry belongs to a copy of $Q$. For $Q$-forcing matrices, we establish the existence and uniqueness of extremal constructions minimizing the number of $1$-entries, characterize them using Young diagrams and corner functions, and derive explicit formulas and monotonicity results. For strongly $Q$-forcing matrices, we show that the minimum possible number of $0$-entries of an $m\times n$ strongly $Q$-forcing matrix is always $O(m+n)$, determine the maximum possible number of $1$-entries of an $n\times n$ strongly $P$-forcing matrix for every $2\times2$ and $3\times3$ permutation matrix, and identify symmetry classes with identical extremal behavior. We further propose a conjectural formula for the maximum possible number of $1$-entries of an $n\times n$ strongly $I_k$-forcing matrix, supported by results for $k=2,3$. These findings reveal contrasting extremal structures between forcing and strongly forcing, extending the combinatorial understanding of pattern embedding in $(0,1)$-matrices.