Enumerating Partial Latin Rectangles

  • Raúl M. Falcón
  • Rebecca J. Stones


This paper deals with different computational methods to enumerate the set $\mathrm{PLR}(r,s,n;m)$ of $r \times s$ 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 given by a symmetric polynomial of degree $3m$, and we determine the leading terms (the monomials of degree $3m$ through $3m-9$) using inclusion-exclusion. For $m \leqslant 13$, exact formulas for these symmetric polynomials are determined using a chromatic polynomial method. Adapting Sade's method for enumerating Latin squares, we compute the exact size of $\mathrm{PLR}(r,s,n;m)$, for all $r \leqslant s \leqslant n \leqslant 7$, and all $r \leqslant s \leqslant 6$ when $n=8$. Using an algebraic geometry method together with Burnside's Lemma, we enumerate isomorphism, isotopism, and main classes when $r \leqslant s \leqslant n \leqslant 6$. Numerical results have been cross-checked where possible.

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