Automatic Counting of Generalized Latin Rectangles and Trapezoids

By George Spahn and Doron Zeilberger

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Written: Aug. 2021

Appeared in Enumerative Combinatorics and Applications Vol. 2: issue 1 (2022) Article S2R8

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.

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Sample Input and Output for LatinTrapezoids.txt

Sample Input and Output for GenLatinRecs.txt

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