Gröbner bases and the number of Latin squares related to autotopisms of order ≤7

Ganfornina, Raúl Manuel Falcón; Martín-Morales, Jorge

doi:10.1016/j.jsc.2007.07.004

Cited by 27 publications

(34 citation statements)

References 4 publications

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“…When α has the cycle structure 3 + 3, we find that Λ (α) = 648 disagrees with the value given in [4], which gave 1296 instead. We can be sure the value here is correct, otherwise (4) would be invalid.…”

Section: Autotopisms and Automorphismscontrasting

confidence: 85%

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The parity of the number of quasigroups

Stones

2010

Discrete Mathematics

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Section: Autotopisms and Automorphismscontrasting

confidence: 85%

“…Therefore the sum (3) becomes I(6) = 1 128 960 + 0 + 1280 + 0 + 144 + 0 + 36 + 96 + 0 + 15 + 0 = 1 130 531. (4) Observe that all of the summands in (4) are integral, and only one of the summands in (4) is odd. This motivates the subsequent study of the divisors of Λ(α).…”

Section: Autotopisms and Automorphismsmentioning

confidence: 99%

The parity of the number of quasigroups

Stones

2010

Discrete Mathematics

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“…Nevertheless, the equivalent problems for partial Latin rectangles have not been dealt with in depth yet. Particularly, by means of computational algebraic geometry, it is known() the number of partial Latin squares for order up to 6 and their distribution into isotopism and isomorphism classes, together with the cardinality of

R_{r, s, n; m}

for r , s , n ≤4 (see the literature() for previous studies about how to use this computational method to deal with Latin squares).…”

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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“…However, there does not exist a similar study for self-orthogonal partial Latin rectangles of any order. In the current paper, we deal with this problem by adapting the Combinatorial Nullstellensatz of Alon [1], whose effectiveness in the study of Latin squares has been exposed in [11,12].…”

Section: Introductionmentioning

confidence: 99%

Enumeration and classification of self-orthogonal partial Latin rectangles by using the polynomial method

Ganfornina

2013

The Seventh European Conference on Combinatorics, Graph Theory and Applications

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The current paper deals with the enumeration and classification of the set SOR r,n of self-orthogonal r × r partial Latin rectangles based on n symbols. These combinatorial objects are identified with the independent sets of a Hamming graph and with the zeros of a radical zero-dimensional ideal of polynomials, whose reduced Gröbner basis and Hilbert series can be computed to determine explicitly the set SOR r,n . In particular, the cardinality of this set is shown for r ≤ 4 and n ≤ 9 and several formulas on the cardinality of SOR r,n are exposed, for r ≤ 3. The distribution of r × s partial Latin rectangles based on n symbols according to their size is also obtained, for all r, s, n ≤ 4.

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Gröbner bases and the number of Latin squares related to autotopisms of order ≤7

Cited by 27 publications

References 4 publications

The parity of the number of quasigroups

The parity of the number of quasigroups

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

Enumeration and classification of self-orthogonal partial Latin rectangles by using the polynomial method

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