Counting partial Latin rectangles and tridimensional rook placements with multisymmetric functions

Emmanuel Briand

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arXiv · 2607.05214

Counting partial Latin rectangles and tridimensional rook placements with multisymmetric functions

Emmanuel Briand

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Abstract

We generalize Gessel's Formula for the number of Latin rectangles to partial Latin rectangles and non-attacking rook placements in a tridimensional chessboard. We also derive explicit short formulas for the generating series of the numbers of non-attacking rook placements on a chessboard with $2$ or $3$ levels. These series also count partial Latin rectangles with $2$ or $3$ rows. The results are obtained following methods developed by MacMahon and Gessel for counting Latin squares and Latin rectangles, by means of scalar products of multisymmetric functions.

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Emmanuel Briand. 2026-07-06. Counting partial Latin rectangles and tridimensional rook placements with multisymmetric functions. https://arxiv.org/abs/2607.05214

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