Automatic Counting of Generalized Latin Rectangles and Trapezoids

By George Spahn and Doron Zeilberger

.pdf    .tex

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.

# Maple packages

• HorizTilings.txt, a Maple package to weight-count Horizontal Tilings

• LatinTrapezoids.txt, a Maple package for counting Latin Trapezoids of Height 3

• GenLatinRecs.txt, a Maple package for counting Generalized Latin Rectangles with 3 rows and generalized derangements

# Sample Input and Output for LatinTrapezoids.txt

• If you want to see the first 100 terms of the sequence enumerating Latin trapezoids with 3 rows
the input gives the output.

• If you want to see all the Reduced (i.e. the first row is 123..n), Latin Triangles of side-length up to 7
the input gives the output.

• If you want to see all the (i.e. the first row is 123..n) Latin Trapezoids of with three rows and base-size up to 7
the input gives the output.

# Sample Input and Output for GenLatinRecs.txt

• If you want to see a web-book with lots of sequences enumerating permutations with various restrictions
the input gives the output.

• If you want to see a web-book with lots of sequences enumerating generalized 3 by n Latin rectangles (with various restrictions)
the input gives the output.

• If you want to see the first 30 terms of the sequence enumerating 3 by n reduced SUPER LATIN RECTANGLES
the input gives the output.