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) `symbolicdynamicalprogramming' algorithms
for computing many terms of `hard to compute sequences', namely the number of Latin trapezoids, generalized derangements, and generalized threerowed 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 Precursive in n.
Our algorithms are fully implemented in Maple, and generated quite a few terms of such sequences.
Game App
Maple packages

HorizTilings.txt,
a Maple package to weightcount 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 sidelength 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 basesize up to 7
the input gives the
output.
Sample Input and Output for GenLatinRecs.txt

If you want to see a webbook with lots of sequences enumerating permutations with various restrictions
the input gives the
output.

If you want to see a webbook 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.
Doron Zeilberger's Home Page