On Computing the Number of Latin Rectangles

Stones, Rebecca J.; Lin, Sheng Hsien; Liu, Xiaoguang; Wang, Gang

doi:10.1007/s00373-015-1643-1

Cited by 3 publications

(4 citation statements)

References 34 publications

(54 reference statements)

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“…The authors have made efforts to ensure the numbers and formulas presented here are as bug-free as possible; we document these efforts in this section. First, notice that some results can immediately be checked by taking into account the already known results mentioned in the introductory section, particularly those ones concerning Latin rectangles [41,50,53,61,62,70] and partial Latin rectangles of small orders [1,20,27,28,30,34,73]. Further, the various source codes used and their output are available from [35].…”

Section: Verificationmentioning

confidence: 99%

“…It is unrealistic to expect a succinct solution to both problems for arbitrary r, s, n, and m, since they include in particular the number of Latin squares of given order n, which is a long-standing research problem in combinatorics. This is known only for order n 11 [41,53]; see [61,62,70] for some related results on Latin rectangles. Currently, the number of partial Latin rectangles is known only for r, s, n 6 [27,28,30].…”

Section: Introductionmentioning

confidence: 99%

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Enumerating Partial Latin Rectangles

Ganfornina

Stones

2020

Electron. J. Combin.

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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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Section: Verificationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Enumerating Partial Latin Rectangles

Ganfornina

Stones

2020

Electron. J. Combin.

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“…Counting, enumerating, and classifying Latin rectangles are classical problems in combinatorial design theory. Currently, it is known() the number of Latin squares of order up to 11 and their distribution into isotopism, isomorphism, and main classes, together with the number of r × s Latin rectangles based on [ n ], for r ≤ s = n ≤11 and some results for r ≤6 and s = n >11(see the literature() and the references therein). Nevertheless, the equivalent problems for partial Latin rectangles have not been dealt with in depth yet.…”

Section: Introductionmentioning

confidence: 99%

Counting and enumerating partial Latin rectangles by means of computer algebra systems and CSP solvers

Ganfornina

Falcón

Núñez

2018

Math Methods in App Sciences

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This paper provides an in‐depth analysis of how computer algebra systems and CSP solvers can be used to deal with the problem of enumerating and distributing the set of r×s partial Latin rectangles based on n symbols according to their weight, shape, type, or structure. The computation of Hilbert functions and triangular systems of radical ideals enables us to solve this problem for all r,s,n≤6. As a by‐product, explicit formulas are determined for the number of partial Latin rectangles of weight up to 6. Further, to illustrate the effectiveness of the computational method, we focus on the enumeration of 3 subsets: (1) noncompressible and regular, (2) totally symmetric, and (3) totally conjugate orthogonal partial Latin squares. In particular, the former enables us to enumerate the set of seminets of point rank up to 8 and to prove the existence of 2 new configurations of point rank 8. Finally, as an illustrative application, it is also exposed a method to construct totally symmetric partial Latin squares that gives rise, under certain conditions, to new families of Lie partial quasigroup rings.

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“…It is unrealistic to expect a succinct solution to both problems for arbitrary r, s, n, and m, since they include in particular the number of Latin squares of given order n, which is a long-standing research problem in combinatorics. This is known only for order n ≤ 11 [38,50]; see [58,59,66] for some related results on Latin rectangles. Currently, the number of partial Latin rectangles is known only for r, s, n ≤ 6 [25,26,27].…”

Section: Introductionmentioning

confidence: 99%

Enumerating partial Latin rectangles

Falcón,

Stones

2019

Preprint

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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 enumerating 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.

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On Computing the Number of Latin Rectangles

Cited by 3 publications

References 34 publications

Enumerating Partial Latin Rectangles

Enumerating Partial Latin Rectangles

Counting and enumerating partial Latin rectangles by means of computer algebra systems and CSP solvers

Enumerating partial Latin rectangles

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