Automatic Counting of Generalized Latin Rectangles and Trapezoids

TL;DR

This paper presents symbolic-dynamical-programming algorithms implemented in Maple for automated enumeration of complex combinatorial structures like Latin trapezoids and generalized Latin rectangles, extending Gessel's theorem.

Contribution

It introduces novel algorithms for computing terms of difficult sequences related to Latin structures and generalizes Gessel's theorem on Latin rectangles.

Findings

Developed Maple algorithms for enumeration of Latin trapezoids and rectangles.

Generated extensive sequence data for complex combinatorial objects.

Proved a generalization of Gessel's theorem on P-recursiveness of Latin rectangle counts.

Abstract

In this case study in ``fully automated enumeration'', we illustrate how to take full advantage of symbolic computation by developing (what we call) `symbolic-dynamical-programming' algorithms for computing many terms of `hard to compute sequences', namely the number of Latin trapezoids, generalized derangements, and generalized three-rowed Latin rectangles. At the end we also sketch the proof of a generalization of Ira Gessel's 1987 theorem that says that for any number of rows, k, the number of Latin rectangles with k rows and n columns is P-recursive in n. Our algorithms are fully implemented in Maple, and generated quite a few terms of such sequences.