Enumerating partial Latin rectangles [article]

Raúl M. Falcón, Rebecca J. Stones
2019 arXiv   pre-print
This paper deals with distinct computational methods to enumerate the set PLR(r,s,n;m) of r × s partial Latin rectangles on n symbols with m non-empty cells. For fixed r, s, and n, we prove that the size of this set is a symmetric polynomial of degree 3m, and we determine the leading terms (the monomials of degree 3m through 3m-9) using inclusion-exclusion. For m ≤ 13, exact formulas for these symmetric polynomials are determined using a chromatic polynomial method. Adapting Sade's method for
more » ... umerating Latin squares, we compute the exact size of PLR(r,s,n;m), for all r ≤ s ≤ n ≤ 7, and all r ≤ s ≤ 6 when n=8. Using an algebraic geometry method together with Burnside's Lemma, we enumerate isomorphism, isotopism, and main classes when r ≤ s ≤ n ≤ 6. Numerical results have been cross-checked where possible.
arXiv:1908.10610v1 fatcat:hkvg24wbknci3mvcuhfbavqtiq