Fuzzy latin squares and balanced permutation pattern statistics

A latin square of order $n$ can be viewed as a partition of the $n \times n$ all-ones matrix into permutation matrix summands. Here, we consider a relaxation in which the matrix summands are allowed to be induced from shorter permutations. For $\sigma \in S_k$, the `fuzzy permutation matrix' $P_\sigma^{\uparrow n}$ arises from combining all $\binom{n}{k}^2$ order-preserving embeddings of the $k \times k$ permutation matrix $P_\sigma$ into an $n \times n$ matrix. We define a fuzzy latin square as a linear combination of $n \times n$ fuzzy permutation matrices $P_\sigma^{\uparrow n}$ equaling a constant matrix. We study various aspects of these objects, including certain relevant vector space dimensions and a census of fuzzy latin squares with a small number of terms. In particular, we determine strong conditions on four-term fuzzy latin squares in the `vanishing' case (when the constant matrix is all zeros). We also report on a computer-assisted classification of six-term fuzzy latin squares in the non-vanishing case.