Counting partial Latin rectangles and tridimensional rook placements with multisymmetric polynomials

Emmanuel Briand

Counting partial Latin rectangles and tridimensional rook placements with multisymmetric polynomials

Combinatorics 2026-07-06 v1

Authors: Emmanuel Briand

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.

Cite

@article{arxiv.2607.05214,
  title  = {Counting partial Latin rectangles and tridimensional rook placements with multisymmetric polynomials},
  author = {Emmanuel Briand},
  journal= {arXiv preprint arXiv:2607.05214},
  year   = {2026}
}