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

Researh Artile Mathematial Methods in the Applied Sienes Reeived XXXX (www.intersiene.wiley.om) DOI: 10.1002/sim.0000 MOS subjet lassiation: 05B15; 13F20; 20N05; 05B25. Counting and enumerating partial Latin retangles by means of omputer algebra systems and CSP solvers Raul M. Falon a  ,  Osar J. Falon b and Juan Nu~nez b This pap er provides an in-depth analysis of how omputer algebra systems and CSP solvers an b e used to deal with the problem of enumerating and distributing the set of r  s partial Latin retangles based on n symb ols aording to their weight, shap e, typ e or struture. The omputation of Hilb ert funtions and triangular systems of radial ideals enables us to solve this problem for all r ; s ; n  6 . As a by-pro dut, expliit formulas are determined for the numb er of partial Latin retangles of weight up to six. Further, in order to illustrate the eetiveness of the omputational metho d, we fo us on the enumeration of three subsets: (a) non-ompressible and regular, (b) totally symmetri, and () totally onjugate orthogonal partial Latin squares. In partiular, the former enables us to enumerate the set of seminets of p oint rank up to eight and to prove the existene of two new ongurations of point rank eight. Finally, as an illustrative appliation, it is also exp osed a metho d to onstrut totally symmetri partial Latin squares that gives rise, under ertain onditions, to new families of Lie partial quasigroup rings. Copyright   2017 John Wiley & Sons, Ltd. Keywords: Partial Latin square; polinomial ring; ideal; seminet; onjugay; orthogonality. 1. Intro dution An r  s partial Latin retangle based on the set [ n ℄ := f 1 ; : : : ; n g is an r  s array in whih eah ell is either empty or ontains one symbol hosen from the set [ n ℄, suh that eah symbol ours at most one in eah row and in eah olumn. Its weight is the number of non-empty ells. This is a Latin retangle if there are not empty ells. If r = s = n , then it is a partial Latin square of order n (a Latin square if there are not empty ells). Hereafter, R r ;s ;n and R r ;s ;n ; m denote, respetively, the set of r  s partial Latin retangles based on [ n ℄ and its subset of elements of weight m . Counting, enumerating and lassifying Latin retangles are lassial problems in ombinatorial design theory. Currently, it is known [1{4℄ the number of Latin squares of order up to 11 and their distribution into isotopism, isomorphism and main lasses, together with the number of r  s Latin retangles based on [ n ℄, for r  s = n  11 and some results for r  6 and s = n > 11 (see [5,6℄ and the referenes therein). Nevertheless, the equivalent problems for partial Latin retangles have not been dealt with in depth yet. Partiularly, by means of omputational algebrai geometry, it is known [7{9℄ the number of partial Latin squares for order up to six and their distribution into isotopism and isomorphism lasses, together with the ardinality of R r ;s ;n ; m for r ; s ; n  4 (see [10,11 ℄ for previous studies about how to use this omputational method in order to deal with Latin squares). b Faulty of Mathematis, Department of Geometry and Topology, University of Seville, / Tara s/n. 41012-Sevilla. a University of Seville, Department of Applied Mathematis I.  Correspondene to: Shool of Building Engineering, University of Seville. Avda. Reina Meredes 4 A, 41012, Seville, Spain. E-mail: rafalganus.es "This is the pre-peer reviewed version of the following article: [R. M. Falcón, O. J. Falcón and J. Núñez. Counting and enumerating partial Latin rectangles by means of computer algebra systems and CSP solvers. Mathematical Methods in the Applied Sciences (2018). DOI: 10.1002/mma.4820], which has been published in final form at [https://doi.org/10.1002/mma.4820]. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving." Mathematial Methods in the Applied Sienes R. M. Falon, O. J. Falon, J. Nu~nez This paper provides an in-depth analysis of how omputational algebrai geometry an be used to enumerate and lassify partial Latin retangles aording not only to their weight, but also to their shape, type and struture. In order to illustrate the eetiveness of this omputational method, we fous on the enumeration of (a) non-ompressible and regular, (b) totally symmetri, and () totally onjugate orthogonal partial Latin squares. The former enables us to deal with the enumeration of seminets (a type of inident struture introdued by Usan [12℄ as a natural generalization of nets), whereas the study of the other two types of partial Latin squares are related to algebrai properties of partial quasigroups (a brief sketh of this study has reently been exposed by the authors in [13℄). Reall in this last regard that a quasigroup of order n [14℄ is a pair ( S;  ) formed by a nite set S of n elements that is endowed with a produt  so that, if any two of the three symbols in the equation a  b =  are given as elements of S , then the third one is uniquely determined. This onept is straightforwardly generalized to that of partial quasigroup of order n , for whih (a) the law  is a partial binary operation, and (b) if both equations a  x = b and y  a = b , with a; b 2 S , have solutions for x ; y 2 S , then both solutions are unique. The multipliation table of a (partial) quasigroup of order n onstitutes indeed a (partial) Latin square of the same order. Bruk [15℄ introdued the onept of totally symmetri quasigroup as a quasigroup ( S;  ) for whih the equation a  b =  remains valid under every permutation of the three symbols a; b ;  2 S . There exist six suh permutations and eah one of them gives rise to a new quasigroup, whih is said to be onjugate to ( S;  ). Hene, a quasigroup is totally symmetri if its six onjugates oinide. If besides, the quasigroup is idempotent , that is, if a  a = a , for all a 2 S , then this notion is equivalent to that of a Steiner triple system . The distribution of totally symmetri quasigroups and Steiner triple systems into isomorphism lasses is known [16,17℄ for orders up to 10 and 19, respetively. Two quasigroups of order n are said to be orthogonal if the juxtaposition of their orresponding multipliation tables gives rise to an n  n array ontaining n 2 distint ordered pairs. Stein [18℄ posed the problem of onstruting a quasigroup or Latin square that is orthogonal to one of its onjugates. In this regard, it is known [19{22℄ the existene of quasigroups that are orthogonal to the onjugate under onsideration, whih is in turn distint from the former, for any order n 62 f 2 ; 3 ; 6 g . Muh more reently, Bennett and Zhang [23℄ dealt with Latin squares for whih eah one of their onjugates is orthogonal to its transpose. They proved the existene of suh Latin squares for all prime powers n 62 f 2 ; 3 ; 5 g . Further, Lindner et al. [24 ℄ foused on idempotent Latin squares for whih their six onjugates are distint and pairwise orthogonal. They proved in partiular the existene of suh Latin squares for every order being a prime power n  8 and also for all suÆiently large orders n . Bennett [25 ℄ established n > 5594 as an upper bound for this last ondition exept possibly n = 6810, and enumerated a series of smaller orders for whih these Latin squares also exist. Four years later, he improved [26℄ the previous upper bound to n > 5074. Muh more reently, Belyavskaya and Popovih [27℄ introdued the equivalent notion of totally onjugate orthogonal quasigroup as a quasigroup for whih its six onjugates are distint and pairwise orthogonal. They proved the existene of suh quasigroups for any order n  11 that is relatively prime to 2, 3, 5, and 7. Their motivation to study this kind of quasigroups was mainly based on their appliation in error deteting odes [28℄. Sine Evans [29℄ introdued the problem of embedding a partial quasigroup of order n into a quasigroup of order 2 n , a wide amount of authors have dealt with the embedding of distint types of partial quasigroups; partiularly, that of a partial totally symmetri quasigroup into a totally symmetri quasigroup [30 {33℄. Further, the orthogonality among onjugates of a partial Latin square was indiretly ontemplated [34{36℄ by fousing on the existene of inomplete Latin squares that are orthogonal to one of their onjugates and have an empty subsquare that an be lled by means of a Latin square that is orthogonal in turn to its orresponding onjugate. A more general ase was proposed by the rst author [8℄, who makes use of omputational algebrai geometry to enumerate the set of self-orthogonal partial Latin squares of order n  4. This paper delves into this topi by dealing with the sets of partial Latin squares of a given order for whih their six onjugates either oinide or are all of them distint and pairwise orthogonal, respetively. In order to improve the omputational eÆieny, it is proposed to fous on tehniques to solve Boolean satisability problems instead of those on algebrai geometry. As an illustrative appliation of the exposed study, we also delve into a reent work developed by the authors [37℄ about the enumeration of partial quasigroup rings over nite elds derived from partial Latin squares. Bruk [15℄ introdued the onept of quasigroup ring related to a quasigroup ( S;  ) as an algebra of basis f e a j a 2 S g over a base eld K suh that e a e b = e a  b , for all a; b 2 S . This onept is straightforwardly generalized to that of partial quasigroup ring in ase of being the pair ( S;  ) a partial quasigroup. In this paper, we desribe a totally symmetri partial Latin square of order 3 n , derived from a given partial Latin square of order n , that enables us to introdue in turn a Lie partial quasigroup ring over a nite eld of harateristi two. 2Copyright   2017 John Wiley & Sons, Ltd. Math. Meth. Appl. Si. 2017 ,001{25 Prepared using mmaauth.ls R. M. Falon, O. J. Falon, J. Nu~nez Mathematial Methods in the Applied Sienes The paper is organized as follows. Setion 2 deals with some preliminary onepts and results on partial Latin squares, seminets and omputational algebrai geometry that are used throughout our study. These results are implemented in Setion 3 to determine the ardinality of R r ;s ;n ; m , for all r ; s ; n  6. In Setion 4, the distribution of non-empty ells per row and olumn and the number of ourrenes of eah symbol enable us to use omputational algebrai geometry in order to identify the set of partial Latin retangles of a given shape, type or struture. The distribution of R r ;s ;n into isotopism and main lasses is then determined for all r ; s ; n  6. As a by-produt, we establish expliit formulas for the number of partial Latin retangles of any order and weight up to six. Setion 5 deals with the distribution into main lasses of seminets of point rank up to eight. We also prove the existene of two new ongurations of seminets with point rank eight that omplete the lassiation given by Lyakh [38℄. In Setion 6, we introdue a pair of series of binary onstraints that haraterize, respetively, the sets of totally symmetri and totally onjugate orthogonal partial Latin squares of given order and weight. Finally, Setion 7 deals with an illustrative method to onstrut a family of Lie partial quasigroup rings from ertain totally symmetri partial Latin squares. 2. Preliminaries This setion deals with some basi results on partial Latin retangles, seminets and omputational algebrai geometry that are used throughout the paper. For more details about these topis, we refer the reader to [12,39 ,40 ℄. 2.1. Partial Latin retangles An entry of a partial Latin retangle P 2 R r ;s ;n is any triple ( i ; j ; k ) 2 [ r ℄  [ s ℄  [ n ℄ that is uniquely related to a non-empty ell of P whih is situated in the i t h row and j t h olumn and ontains the symbol k . The partial Latin retangle P is uniquely determined by the set of all its entries, whih is denoted as E ( P ). Thus, for instane, the partial Latin square P in Figure 1 belongs to the set R 3 ; 3 ; 3;4 and has f (1 ; 1 ; 2) ; (1 ; 2 ; 1) ; (2 ; 1 ; 1) ; (3 ; 3 ; 3) g as set of entries. P  2 1 1 3 Q  1 3 2 3 Figure 1. Isotopi partial Latin squares in R 3 ; 3 ; 3;4 . Let S m denote the symmetri group on m elements. An isotopism of R r ;s ;n is any triple  = ( ;  ;  ) 2 S r  S s  S n , where  ,  and  onstitute, respetively, a permutation of the rows, olumns and symbols of any partial Latin retangle P 2 R r ;s ;n . This gives rise to the isotopi partial Latin retangle P  2 R r ;s ;n , whose set of entries is E ( P  ) = f (  ( i ) ;  ( j ) ;  ( k )) : ( i ; j ; k ) 2 E ( P ) g . Thus, for instane, both partial Latin squares in Figure 1are isotopi by means of the isotopism ((123) ; (12) ; (13)). Permutations among the three omponents of all the entries of a partial Latin retangle also give rise to new partial Latin retangles. In this regard, let  be a permutation in S 3 . The  -onjugate of P 2 R r ;s ;n is dened as the partial Latin retangle P  having as set of entries the set E ( P  ) = f ( p  (1) ; p  (2) ; p  (3) ) : ( p 1 ; p 2 ; p 3 ) 2 E ( P ) g . If the permutation  preserves the set R r ;s ;n , then  is said to be a parastrophism . Hene, the set of parastrophisms of R r ;s ;n is  f Id g if r , s and n are pairwise distint.  f Id ; (12) g if r = s 6 = n .  f Id ; (13) g if r = n 6 = s .  f Id ; (23) g if s = n 6 = r .  S 3 if r = s = n . There are, therefore, six onjugates: P Id = P , P (12) = P t , P (13) , P (23) , P (123) = ( P (23) ) t and P (132) = ( P (13) ) t ; where t denotes the transpose of the orresponding partial Latin retangle. Figure 2shows, for instane, a partial Latin square P whose six onjugates are pairwise distint. The partial Latin square P that is shown in Figure 1is, however, an example for whih all its six onjugates oinide. Suh a partial Latin square is said to be totally symmetri . Hereafter, we denote respetively as TS n and TS n ; m the set of totally symmetri partial Latin squares of order n and its subset of partial Latin squares of weight m . Math. Meth. Appl. Si. 2017, 00 1{25 Copyright   2017 John Wiley & Sons, Ltd. 3 Prepared using mmaauth.ls Mathematial Methods in the Applied Sienes R. M. Falon, O. J. Falon, J. Nu~nez P  1 2 3 1 P t  1 2 3 1 P (13)  1 3 1 2 P (23)  1 2 2 3 P (123)  1 3 2 2 P (132)  1 1 2 3 Figure 2. Partial Latin square in R 3 ; 3 ; 3;4 and its onjugates. Two partial Latin retangles are said to be paratopi if one of them is isotopi to a onjugate of the other. To be isotopi, parastrophi or paratopi are equivalene relations among partial Latin retangles. They make possible the respetive distribution of partial Latin retangles into isotopism , parastrophism and main lasses. A partial Latin square P of order n is said to be non-ompressible if this does not ontain empty rows or empty olumns, or if all the n symbols appear as entries in E ( P ). This is said to be regular if: (a) there does not exist a ell that is, simultaneously, the only non-empty ell in its row and its olumn, and (b) any row or olumn with exatly one non-empty ell ontains a symbol that appears at least twie in E ( P ). Thus, for instane, the partial Latin square P in Figure 2is non-ompressible. Nevertheless, it is not regular, beause: (a) both its third row and its third olumn have exatly one non-empty ell, whih is ommon to both of them, and (b) its seond row ontains exatly one non-empty ell, but the symbol therein only appears one in P . Two partial Latin squares of order n , P = ( p i j ) and Q = ( q i j ), are said to be orthogonal if all the ordered pairs on nonempty entries that are obtained when both arrays are superimposed are distint. Equivalently, given i ; i 0 ; j ; j 0 2 [ n ℄ suh that p i j = p i 0 j 0 2 [ n ℄, then q i j and q i 0 j 0 are not the same symbol of [ n ℄. Thus, for instane, the partial Latin squares P and P (13) in Figure 2are orthogonal, but the partial Latin squares P and P (12) in the same gure are not. Now, let us onsider a non-trivial permutation  2 S 3 n f Id g . A partial Latin square P 2 R n;n ;n is said to be  -orthogonal if it is orthogonal to its  -onjugate. This is self-orthogonal if  = (12). Thus, for instane, the partial Latin square P (23) in Figure 2is self-orthogonal. Further, we say that a partial Latin square is totally onjugate orthogonal if its six onjugates are distint and pairwise orthogonal. This is the ase, for instane, of the partial Latin square in Figure 3. From here on, the set of totally onjugate orthogonal partial Latin squares of order n and its subset of partial Latin squares of weight m are respetively denoted as TCO n and TCO n ; m . P  3 2 1 3 P t  1 3 3 2 P (13)  3 2 3 1 P (23)  3 3 1 2 P (123)  1 3 3 2 P (132)  3 3 2 1 Figure 3. Totally onjugate orthogonal partial Latin square in R 3 ; 3 ; 3;4 . 2.2. Seminets Bates [41℄ dened a halfnet as an inidene struture of points and lines suh that: (a) there exist three distint parallel lasses of lines, (b) every point is on at most one line of eah lass, and () any two lines belonging to distint lasses meet in at most one point. The number of points onstitutes the point rank of a halfnet. Two halfnets are in the same isomorphism lass if there exists a permutation among the points that preserves ollinearity in eah parallel lass. If this happens after relabeling their parallel lasses, then they are in the same main lass . Currently, the distribution of halfnets into isomorphism and main lasses is only partially known for nets and, to a muh lesser extent, seminets. Bruk [42℄ dened a net of order n as a halfnet of n 2 points and 3 n lines in whih every point is on exatly one line of eah parallel lass, any two lines from distint parallel lasses meet in exatly one point and there exists at least one line with exatly n distint points. Hene, every line ontains n points and every parallel lass is formed by n lines. More reently and motivated by its appliation in oding theory, Usan [12℄ introdued the onept of seminet as a halfnet in whih every point is on exatly one line of eah parallel lass and any two lines meet in at most one point. Unlike nets, the lines of a seminet an ontain dierent numbers of points and its parallel lasses an have dierent numbers of lines. The L -order of a seminet is the maximum number of lines in a parallel lass. If all the lines have the same number n of points, then all the parallel lasses have the same number m of lines. In this ase, the seminet is said to be n -regular . If, furthermore, m = n , then it is a net of order n . 4Copyright   2017 John Wiley & Sons, Ltd. Math. Meth. Appl. Si. 2017 ,001{25 Prepared using mmaauth.ls R. M. Falon, O. J. Falon, J. Nu~nez Mathematial Methods in the Applied Sienes  1 2 3 4 2 1 4 3 3 4 1 2 4 3 2 1 Figure 4. Net identied with a Latin square of order 4. Every net of order n an be identied with a Latin square of the same order. The points and parallel lasses of the net are respetively identied with the ells of the Latin square and its sets of ells sharing the same row, olumn or symbol (see Figure 4). In addition, Stojakovi and Usan [43 ℄ proved that every seminet of L -order n an be identied with a non-ompressible regular partial Latin square of order n in a similar way that nets do with Latin squares. In this ase, the points of the seminet are identied with the non-empty ells of the partial Latin square (see Figure 5). As a onsequene, the distribution of nets and seminets into isomorphism and main lasses results, respetively, from the equivalent distribution of Latin squares and non-ompressible regular partial Latin squares into isotopism and main lasses.  1 2 1 2 2 Figure 5. Seminet identied with a partial Latin square of order 4 and weight 5. Havel [44℄ dened a onguration as a seminet ontaining at least four points suh that every line ontains at least two points and any two points P and Q of the seminet are onneted , that is to say, there exists a sequene of points and lines, P 0 ; l 0 ; P 1 ; l 1 ;:::;P m , suh that P 0 = P , P m = Q and eah pair of points P i  1 and P i are on the line l i  1 , for all i  m . Havel determined the main lasses of those ongurations with point rank up to seven and, shortly after, Lyakh [38℄ gave a lassiation of those ongurations with point rank eight. 2.3. Computational algebrai geometry Let X and K [ X ℄ respetively be the ordered set of n variables f x 1 ;:::;x n g and the related multivariate polynomial ring K [ x 1 ;:::;x n ℄ over a base eld K . The lass of a polynomial p 2 K [ X ℄ is the minimum i  n suh that p 2 K [ x 1 ;:::;x i ℄. A triangular system in K [ X ℄ is a nite ordered set of polynomials f p 1 ;:::;p m g  K [ X ℄ suh that the lass of p i is less than the lass of p i +1 , for all i < m . An ideal of polynomials in K [ X ℄ is any subset I  K [ X ℄ suh that 0 2 I ; p + q 2 I , for all p ; q 2 I ; and p q 2 I for all p 2 I and q 2 K [ X ℄. A subideal of I is any subset J  I that is also an ideal in K [ X ℄. The ideal generated by a nite set of polynomials f p 1 ;:::;p m g  K [ X ℄ is dened as the set f q 1 p 1 + ::: + q n p n : q 1 ;:::;q n 2 K [ X ℄ g . The aÆne variety V ( I ) is the set of points in K n that are zeros of all the polynomials in I . If this is nite, then the ideal I is zero-dimensional . It is radial if it ontains all the polynomials p 2 K [ X ℄ so that p m 2 I for some natural m . A term order on the set of monomials of K [ X ℄ is a multipliative well-ordering whose smallest element is the onstant monomial 1. Thus, for instane, the lexiographi term order < lex is dened so that, given two monomials X a = x a 1 1 : : : x a n n and X b = x b 1 1 : : : x b n n , one has that X a < lex X b if there exists a natural m  n suh that a i = b i for all i  m and a m < b m . The largest monomial of a polynomial with respet to a term order is its leading monomial . The initial ideal of an ideal I  K [ X ℄ is the ideal generated by the leading monomials of the non-zero polynomials of I . Any subset G  I whose leading monomials generate this initial ideal is alled a Grobner basis of I with respet to the underlying term order. Any monomial of I that is not ontained in its initial ideal is alled standard . Regardless of the monomial term ordering, if the ideal I is zero-dimensional Math. Meth. Appl. Si. 2017, 00 1{25 Copyright   2017 John Wiley & Sons, Ltd. 5 Prepared using mmaauth.ls Mathematial Methods in the Applied Sienes R. M. Falon, O. J. Falon, J. Nu~nez and radial, then the number of standard monomials in I oinides with the Krull dimension of the quotient ring K [ X ℄ =I and with the ardinality of V ( I ). This is obtained by means of the Hilbert funtion , whih maps eah non-negative integer m onto HF K [ X ℄ =I ( m ) = dim K ( K [ X ℄ m = ( K [ X ℄ m \ I )). Here, K [ X ℄ m denotes the set of homogeneous polynomials in K [ X ℄ of degree m and HF K [ X ℄ =I ( m ) oinides with the number of standard monomials in I of degree m . The problem of omputing Hilbert funtions is NP-omplete [45℄. Its omputation is based on that of a Grobner basis of the ideal, whose omplexity in ase of dealing with a zero-dimensional ideal is d O ( n ) [46℄, where d is the maximal degree of the polynomials and n is the number of variables. The next result indiates how omputational algebrai geometry an be used to enumerate and ount the partial Latin retangles in the set R r ;s ;n . Hereafter, the set of variables and the base eld of the polynomial ring to be onsidered are, respetively, X = f x 111 ;:::;x r s n g and the nite eld F 2 . Theorem 2.1 ( [8℄) The set R r ;s ;n is identied with the set of zeros of the zero-dimensional radial ideal in F 2 [ X ℄ I r ;s ;n := h x ijk x i 0 j k ; x ijk x i j 0 k ; x ijk x ijk 0 : i ; i 0  r ; j ; j 0  s ; k ; k 0  n i : Besides, jR r ;s ;n ; m j = HF F 2 [ X ℄ =I r ;s ;n ( m ) ; for all m  0 ; and jR r ;s ;n j = dim F 2 ( F 2 [ X ℄ =I r ;s ;n ) : The proof of Theorem 2.1 is based on the fat that every standard monomial x a 111 111 :::x a r s n r s n of the ideal I r ;s ;n an be identied with a partial Latin retangle in R r ;s ;n with set of entries f ( i ; j ; k ) 2 [ r ℄  [ s ℄  [ n ℄ : a ijk = 1 g . Partiularly, the presene of the monomial x ijk x i 0 j k as generator of the ideal I r ;s ;n involves the non-existene of the symbol k twie in the j t h olumn; that of x ijk x i j 0 k involves the non-existene of the symbol k twie in the i t h row; and that of x ijk x ijk 0 involves the non-existene of two distint symbols in the ell ( i ; j ). Based on this result, the speialized algorithm desribed by Dikenstein and Tobis [47℄ was implemented in [8℄ for omputing the ardinality of R r ;s ;n ; m , for all r ; s ; n  4. For higher orders, however, the required omputational ost turned out to be exessive due to large memory storage requirements. This ost is only due to the omputation of the orresponding Hilbert funtion, beause the set of generators of I r ;s ;n onstitutes itself a lexiographi Grobner basis of the ideal. To redue it, an alternative proedure is introdued in the next setion. This is based on the similarity that exists among those generators in I r ;s ;n that orrespond to distint rows in a partial Latin retangle. A preliminary version of this proedure was exposed in [9℄, where the ardinality of R r ;s ;n was omputed for all r ; s ; n  6. For a better understanding of this proedure, the orresponding omputation of jR 3 ; 3 ; 3;2 j is illustrated in Example 1. 3. An alternative pro edure to ompute jR r ;s ;n j For eah positive integer i  r we dene the zero-dimensional subideal I ( i ) r ;s ;n := h x ijk x i j 0 k ; x ijk x ijk 0 : j ; j 0  s ; k ; k 0  n i  I r ;s ;n : There exist distint algorithms [48{50℄ that enable us to deompose the zero-dimensional ideal I (1) r ;s ;n into a nite set f J 1 ; 1 ;:::;J 1 ;t g of subideals generated by triangular systems and whose aÆne varieties onstitute a partition of V ( I (1) r ;s ;n ). The omplexity of this omputation in the mentioned algorithms is polynomial one a lexiographi Grobner basis of the ideal is known. This is our ase, beause the set of generators of I (1) r ;s ;n onstitutes itself one suh a basis. Now, for eah i > 1 and l  t , let J i ;l be the subideal of I ( i ) r ;s ;n whose generators oinide with those of J 1 ;l after replaing eah variable x 1 j k by x ijk . For eah tuple ( t 1 ;:::;t r ) 2 [ t ℄ r we dene the ideal K t 1 ;::: ;t r := J 1 ;t 1 + : : : + J r ;t r + h x ijk x i 0 j k : i ; i 0  r ; j  s ; k  n i : (1) The triangularity of the underlying systems involves eah subideal J i ;t j to have at least one generator of the form x i j 0 k or x i j 0 k  1. The number of generators of the seond form in the ideal K t 1 ;::: ;t r onstitutes the minimum number of entries in a partial Latin retangle that is identied with a point in V ( K t 1 ;::: ;t r ). We denote this number by m t 1 ;:::;t r . 6Copyright   2017 John Wiley & Sons, Ltd. Math. Meth. Appl. Si. 2017 ,001{25 Prepared using mmaauth.ls R. M. Falon, O. J. Falon, J. Nu~nez Mathematial Methods in the Applied Sienes Prop osition 3.1 Let m be a non-negative integer. Then HF F 2 [ X ℄ =I r ;s ;n ( m ) = X ( t 1 ;:::;t r ) 2 [ t ℄ r m t 1 ;:::;t r  m HF F 2 [ X ℄ =K t 1 ;:::;t r ( m  m t 1 ;::: ;t r ) : Pro of. Let X a = x a 111 111 : : : x a r s n r s n be a standard monomial of degree m in I r ;s ;n . Sine the ideals desribed in (1) onstitute a partition of the aÆne variety V ( I r ;s ;n ), there exists exatly one ideal K t 1 ;:::;t r that ontains the point ( a 111 ;:::;a r s n ) 2 V ( I r ;s ;n ). The result follows then from the fat that the monomial X a is uniquely related to the standard monomial x a 0 111 111 :::x a 0 r s n r s n of degree m  m t 1 ;::: ;t r in K t 1 ;::: ;t r , where a 0 ijk = 0 if x ijk  1 is a generator of K t 1 ;::: ;t r and a 0 ijk = a ijk , otherwise. 2 The smaller number of variables that are required to ompute eah addend in Proposition 3.1, together with the triangularity of the involved system and the possible parallel omputation to determine distint addends at the same time, redue the running time and ost of omputation of HF F 2 [ X ℄ =I r ;s ;n ( m ) in omparison with Theorem 2.1. Moreover, we do not need to ompute all these addends, beause HF F 2 [ X ℄ =K t 1 ;:::;t r ( m ) = HF F 2 [ X ℄ =K t  (1) ;:::;t  ( r ) ( m ), for all ( t 1 ;:::;t r ) 2 [ t ℄ r , m  0 and  2 S r . Example 3.2 The ideal I (1) 3 ; 3 ; 3 related to the rst row of a partial Latin square of order 3 an be deomposed into the next six disjoint subideals i) J 1 ; 1 = I (1) 3 ; 3 ; 3 + h x 111 ; x 121 ; x 131 i . ii) J 1 ; 2 = I (1) 3 ; 3 ; 3 + h x 111 ; x 121 ; x 131  1 ; x 132 ; x 133 i . iii) J 1 ; 3 = I (1) 3 ; 3 ; 3 + h x 111 ; x 121  1 ; x 122 ; x 123 ; x 131 i . iv) J 1 ; 4 = I (1) 3 ; 3 ; 3 + h x 111  1 ; x 112 ; x 113 ; x 121 ; x 122 ; x 131 ; x 132 i . v) J 1 ; 5 = I (1) 3 ; 3 ; 3 + h x 111  1 ; x 112 ; x 113 ; x 121 ; x 122 ; x 131 ; x 132  1 ; x 133 i . vi) J 1 ; 6 = I (1) 3 ; 3 ; 3 + h x 111  1 ; x 112 ; x 113 ; x 121 ; x 122  1 ; x 123 ; x 131 ; x 132 i . Partial Latin squares of order 3 are then distributed as points of 1. V ( J 1 ; 1 ) if they do not ontain the symbol 1 in their rst row. 2. V ( J 1 ; 2 ) if they ontain the symbol 1 in the ell (1 ; 3) . 3. V ( J 1 ; 3 ) if they ontain the symbol 1 in the ell (1 ; 2) . 4. V ( J 1 ; 4 ) if they ontain the symbol 1 in the ell (1 ; 1) but do not ontain the symbol 2 in their rst row. 5. V ( J 1 ; 5 ) if they ontain the symbol 1 in the ell (1 ; 1) and the symbol 2 in the ell (1 ; 3) . 6. V ( J 1 ; 6 ) if they ontain the symbol 1 in the ell (1 ; 1) and the symbol 2 in the ell (1 ; 2) . For eah triple ( t 1 ; t 2 ; t 3 ) 2 [6℄ 3 , we onsider the ideal K t 1 ;t 2 ;t 3 = J 1 ;t 1 + J 2 ;t 2 + J 3 ;t 3 + h x ijk x i 0 j k : i ; i 0 ; j ; k  3 i : The values of HF F 2 [ X ℄ =K t 1 ;t 2 ;t 3 are exposed in Table 1. Let m t 1 ;t 2 ;t 3 be the number of generators of the form x ijk  1 in the ideal K t 1 ;t 2 ;t 3 . Thus, for instane, every point of the aÆne variety V ( K 6 ; 3 ; 2 ) is uniquely related to a partial Latin square of order 3 and weight at least m 6 ; 3 ; 2 = 4 . This last value holds from the fat that the set of entries of any suh a partial Latin square always ontains the subset f (1 ; 1 ; 1) ; (1 ; 2 ; 2) ; (2 ; 2 ; 1) ; (3 ; 3 ; 1) g . From Proposition 3.1, we have, for example, that jR 3 ; 3 ; 3:2 j = HF F 2 [ X ℄ =K 1 ; 1 ; 1 (2) + 3 HF F 2 [ X ℄ =K 1 ; 1 ; 2 (1) + 3 HF F 2 [ X ℄ =K 1 ; 1 ; 3 (1) + 3 HF F 2 [ X ℄ =K 1 ; 1 ; 4 (1) + 3 HF F 2 [ X ℄ =K 1 ; 1 ; 5 (0)+ 3 HF F 2 [ X ℄ =K 1 ; 1 ; 6 (0) + 6 HF F 2 [ X ℄ =K 1 ; 2 ; 3 (0) + 6 HF F 2 [ X ℄ =K 1 ; 2 ; 4 (0) + 6 HF F 2 [ X ℄ =K 1 ; 3 ; 4 (0) = 270 :  Math. Meth. Appl. Si. 2017, 00 1{25 Copyright   2017 John Wiley & Sons, Ltd. 7 Prepared using mmaauth.ls Mathematial Methods in the Applied Sienes R. M. Falon, O. J. Falon, J. Nu~nez Table 1. Hilbert funtions related to the set R 3 ; 3 ; 3 . HF F 2 [ X ℄ =K t 1 ;t 2 ;t 3 ( m ) t 1 :t 2 :t 3 m 1.1.1 1.1.2 1.1.3 1.1.4 1.1.5 1.1.6 1.2.3 1.2.4 1.2.5 1.2.6 1.3.4 1.3.5 1.3.6 2.3.4 2.3.5 2.3.6 01111111111111111 1 18 16 16 14 11 11 14 12 10 9 12 9 10 10 8 8 2 108 84 84 62 36 36 64 45 29 24 45 24 29 32 19 19 3 264 176 176 104 42 42 116 63 29 23 63 23 29 38 16 16 4 270 150 150 66 18 18 84 32 11 8 32 8 11 16 5 5 5 108 48 48 12 2 2 24 5 1 1 5 1 1 2 1 1 612440002000000000 This omputational algebrai method has been implemented in the proedure PLR of the library pls.lib , available online on http://personales.us.es/ raufalgan/LS/pls.lib , for the open omputer algebra system for polynomial omputations Singular [51℄. The orretness and termination of this proedure are based on those of the algorithms desribed in [47,48,50℄ for omputing Hilbert funtions. In order to test its eÆieny, we have rstly heked the known ardinality of R r ;s ;n ; m , for all r ; s ; n  4 (see Table 2), whih was already omputed in [8℄. In the same omputer system, an Intel Core i7-2600 CPU (8 ores), with a 3.4 GHz proessor and 16 GB of RAM , the maximum running time dereases from 50 seonds in [8℄ to less than 1 seond. This orresponds to the omputation of the series jR 4 ; 4 ; 4; m j . The proedure has then been applied for omputing in Tables 3{5the rest of ases so that r  s  n  6. The running time ranges here from less than 1 seond to 32 hours. This maximum running time orresponds to the omputation of the series jR 6 ; 6 ; 6; m j , for whih 2,3 GB of RAM is required. For higher orders, the rst series whose omputation turned out to be exessive for our omputer system due to large memory storage requirements was jR 6 ; 7 ; 7; m j . In order to improve the eÆieny of this omputational algebrai method, we propose in the next setion to impose some extra algebrai onditions to our base ideal. They are referred to the distribution of non-empty ells per row and olumn in a partial Latin retangle and to the number of ourrenes of eah symbol. Table 2. Distribution of R r ;s ;n aording to the weight, for r  s  n  4. jR r ;s ;n ; m j r :s :n m 1.1.1 1.1.2 1.1.3 1.1.4 1.2.2 1.2.3 1.2.4 1.3.3 1.3.4 1.4.4 2.2.2 2.2.3 2.2.4 2.3.3 2.3.4 2.4.4 3.3.3 3.3.4 3.4.4 4.4.4 0111111111111111 1 1 1 1 1 1 1 2 3 4 4 6 8 9 12 16 8 12 16 18 24 32 27 36 48 64 2 2 6 12 18 36 72 16 42 80 108 204 384 270 504 936 1728 3 6 24 96 8 48 144 264 768 2208 1278 3552 9696 25920 4 24 2 18 84 270 1332 6504 3078 13716 58752 239760 5 108 1008 9792 3834 29808 216864 1437696 6 12 264 7104 2412 36216 494064 5728896 7 2112 756 23760 691200 15326208 8 216 108 7776 581688 27534816 9 12 1056 283584 32971008 10 75744 25941504 11 10368 13153536 12 576 4215744 13 847872 14 110592 15 9216 16 576 Total 2 3 4 5 7 13 21 34 73 209 35 121 325 781 3601 28353 11776 116425 2423521 127545137 4. Shap e, typ e and struture of partial Latin retangles The shape of a partial Latin retangle P = ( p i j ) 2 R r ;s ;n is dened as the r  s binary array B P = ( b i j ) suh that b i j = 1 if ( i ; j ; p i j ) 2 E ( P ) and 0, otherwise. Let r i ,  j and s k respetively be the number of lled ells in the i t h row and j t h olumn of P and the number of ourrenes of the symbol k in P . Aording to the terminology exposed by Keedwell [52℄ and generalized by Bean et al. [53℄, the tuples R = (r 1 ;:::; r r ), C = ( 1 ;:::;  s ) and S = (s 1 ;:::; s n ) determine, respetively, the row , olumn and 8Copyright   2017 John Wiley & Sons, Ltd. Math. Meth. Appl. Si. 2017 ,001{25 Prepared using mmaauth.ls R. M. Falon, O. J. Falon, J. Nu~nez Mathematial Methods in the Applied Sienes Table 3. Distribution of R r ;s ; 5 aording to the weight, for r  s  5. jR r ;s ; 5; m j r :s : 5 m 1.1.5 1.2.5 1.3.5 1.4.5 1.5.5 2.2.5 2.3.5 2.4.5 2.5.5 3.3.5 3.4.5 3.5.5 4.4.5 4.5.5 5.5.5 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5 10 15 20 25 20 30 40 50 45 60 75 80 100 125 2 20 60 120 200 130 330 620 1000 810 1500 2400 2760 4400 7000 3 60 240 600 320 1680 4800 10400 7590 20520 43200 54240 112800 233000 4 120 600 260 4140 20040 61400 40500 169920 486000 676200 1881600 5159000 5 120 4680 45600 211440 126900 891360 3594960 5641920 21612480 80602200 6 1920 54480 421200 232680 3018000 17930400 32423520 176546400 920160000 7 30720 465600 240840 6605280 60912000 130248960 1045147200 7845192000 8 6360 262200 128520 9224280 140826600 367731360 4530640800 50648616000 9 63600 27480 7983840 219307800 728440320 14444083200 249687408000 10 5280 4063680 225419040 1004380800 33852910080 944069668800 11 1100160 148010400 950238720 58065734400 2741210616000 12 120960 59047200 603722880 72278294400 6104066712000 13 13284000 249580800 64484985600 10385299320000 14 1512000 63884160 40544726400 13420351008000 15 66240 9216000 17571260160 13065814483200 16 590400 5099169600 9486099648000 17 953107200 5073056640000 18 108288000 1970474400000 19 6681600 547608096000 20 161280 107330054400 21 14667552000 22 1388160000 23 91008000 24 4032000 25 161280 Total 6 31 136 501 1546 731 12781 162661 1502171 805366 33199561 890442316 4146833121 313185347701 64170718937006 Table 4. Distribution of R r ;s ; 6 aording to the weight, for r  s  6 (I). jR r ;s ; 6; m j r :s : 6 m 1.1.6 1.2.6 1.3.6 1.4.6 1.5.6 1.6.6 2.2.6 2.3.6 2.4.6 2.5.6 2.6.6 3.3.6 3.4.6 3.5.6 3.6.6 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6 12 18 24 30 36 24 36 48 60 72 54 72 90 108 2 30 90 180 300 450 192 486 912 1470 2160 1188 2196 3510 5130 3 120 480 1200 2400 600 3120 8880 19200 35400 13896 37344 78360 141840 4 360 1800 5400 630 9990 48060 146700 349650 94770 392580 1115100 2547450 5 720 4320 15120 146880 678240 2168640 389340 2676240 10667160 31419360 6 720 8520 245760 1899600 8546880 961380 12082680 70540800 274470480 7 204480 3139200 21211200 1375920 36270720 326808000 1727352000 8 65160 2881800 32189400 1038960 71633160 1064140200 7893282600 9 1303200 28267200 317760 90585600 2422568400 26212965600 10 222480 13063680 69603840 3803369040 62938898640 11 2669760 29255040 4021099200 108045861120 12 190800 5112000 2756361600 130246779600 13 1152144000 107367120000 14 262828800 58252478400 15 24791040 19683613440 16 3828798720 17 384652800 18 15321600 Total 7 43 229 1045 4051 13327 1447 37273 720181 10291951 108694843 4193269 317651473 15916515301 526905708889 symbol types of P . The type of P is then dened as the triple ( R ; C; S ). Thus, for instane, the type of the partial Latin square of Figure 5is ((2 ; 2 ; 1 ; 0) ; (2 ; 1 ; 1 ; 1) ; (2 ; 3 ; 0 ; 0)). Hereafter, the set of partial Latin retangles of type ( R ; C; S ) is denoted by R R;C;S . Let T n;m be the set of n -tuples T = ( t 1 ;:::;t n ) of weight P i  n t i = m whose omponents are non-negative integers. The onjugate of T is the tuple T  = (t  1 ;:::; t  m ), where eah t  i is the number of positive integers j  n suh that t j  i . If T = (t 1 ;:::; t n ) 2 T n;m is obtained after a dereasing rearrangement of the omponents of T , then T is said to be majorized by a seond tuple T 0 = (t 0 1 ;:::; t 0 n ) 2 T n;m if P i  j t i  P i  j t 0 i , for all j  n . This gives rise to the so-alled dominane order  on T n;m [54℄. Theorem 4.1 Let ( R ; C; S ) 2 T r ;m  T s ;m  T n;m . The set R R;C;S is non-empty only if C  R  , S  C  and R  S  . Math. Meth. Appl. Si. 2017, 00 1{25 Copyright   2017 John Wiley & Sons, Ltd. 9 Prepared using mmaauth.ls Mathematial Methods in the Applied Sienes R. M. Falon, O. J. Falon, J. Nu~nez S3S4,1 S4,2 S4,3 S4,4 S5,1 S5,2 S5,3 S5,4 S5,5 S5,6 S5,7 S6,1 S6,2 S6,3 S6,4 S6,5 S6,6 S6,7 S6,8 S6,9 S6,10 S6,11 S6,12 S6,13 S6,14 S6,15 S6,16 S6,17 S6,18 S6,19 S6,20 S6,21 S6,22 S6,23 S6,24 S6,25 S6,26 S6,27 S6,28 S6,29 S6,30 S6,31 S6,32 S6,33 S6,34 S6,35 S6,36 S6,37 S6,38 S6,39 S6,40 S6,41 S6,42 S6,43 S6,44 S6,45 S6,46 S6,47 S6,48 S6,49 S6,50 S6,51 S6,52 S6,53 S6,54 S6,55 Figure 7. Classiation of seminets with point rank up to six. 16 Copyright   2017 John Wiley & Sons, Ltd. Math. Meth. Appl. Si. 2017 ,001{25 Prepared using mmaauth.ls R. M. Falon, O. J. Falon, J. Nu~nez Mathematial Methods in the Applied Sienes Table 7. Distribution into main lasses of the set R reg R;C;S . m z R z C z S  reg MC 3 21 21 21 1 1 4 2 2 2 2 2 2 2 1 21 2 4 1 1 4 24 1 21 2 21 2 4 1 5 32 2 2 1 2 2 1 4 1 21 3 12 1 31 2 31 2 2 2 1 4 1 2 2 1 2 2 1 8 1 2 2 1 2 2 1 2 2 1 32 2 21 3 24 1 6 42 2 2 1 2 2 2 1 2 8 1 3 2 2 3 2 3 12 1 2 2 1 2 36 2 21 4 144 1 1 6 720 1 2 2 1 2 2 2 1 2 48 2 21 4 48 1 41 2 2 2 1 2 2 2 1 2 16 1 321 321 321 1 1 31 3 6 1 2 3 12 2 2 2 1 2 20 4 21 4 24 1 2 3 2 3 36 1 2 2 1 2 120 5 21 4 288 2 2 2 1 2 2 2 1 2 160 4 2 3 2 3 2 3 144 2 31 3 72 1 2 2 1 2 432 4 21 4 1,296 2 1 6 4,320 1 31 3 31 3 36 1 2 2 1 2 144 2 2 2 1 2 2 2 1 2 624 7 21 4 288 1 2 2 1 2 2 2 1 2 2 2 1 2 160 3 7 43 2 3 1 2 3 1 54 2 2 2 1 3 144 2 21 5 360 1 2 2 1 3 2 2 1 3 144 1 421 321 2 321 2 4 1 2 3 1 36 3 2 2 1 3 48 2 31 4 2 3 1 144 1 2 3 1 2 3 1 162 4 2 2 1 3 360 5 21 5 360 1 2 2 1 3 2 2 1 3 144 1 3 2 1 32 2 32 2 4 1 321 2 12 2 31 4 48 1 2 3 1 72 3 2 2 1 3 192 4 21 5 480 1 321 2 321 2 24 2 31 4 48 1 2 3 1 120 5 2 2 1 3 144 3 2 3 1 2 3 1 612 6 m z R z C z S  reg MC 7 3 2 1 2 3 1 2 2 1 3 1,008 7 21 5 720 1 2 2 1 3 2 2 1 3 288 1 32 2 32 2 32 2 16 3 321 2 48 5 31 4 144 2 2 3 1 192 7 2 2 1 3 720 12 21 5 2,640 5 1 7 10,080 1 321 2 321 2 112 9 31 4 192 2 2 3 1 456 19 2 2 1 3 816 18 21 5 480 1 31 4 2 3 1 1,008 4 2 2 1 3 288 1 2 3 1 2 3 1 1,692 16 2 2 1 3 3,744 26 21 5 6,480 5 2 2 1 3 2 2 1 3 2,592 6 321 2 321 2 321 2 144 5 2 3 1 684 18 2 2 1 3 264 5 31 4 2 3 1 432 2 2 3 1 2 3 1 2,556 21 2 2 1 3 2,088 15 31 4 2 3 1 2 3 1 3,456 3 2 3 1 2 3 1 2 3 1 8,478 13 2 2 1 3 10,152 16 2 2 1 3 2 2 1 3 2,160 3 8 53 2 3 1 2 2 3 1 2 144 1 2 2 1 4 288 1 4 2 2 4 2 4 216 2 2 3 1 2 528 3 2 2 1 4 2,016 3 21 6 8,640 1 1 8 40,320 1 2 3 1 2 2 3 1 2 792 4 2 2 1 4 1,440 3 21 6 1,440 1 2 2 1 4 2 2 1 4 576 1 521 321 3 2 3 1 2 72 1 2 3 1 2 2 3 1 2 432 2 2 2 1 4 576 1 431 32 2 1 32 2 1 24 4 321 3 72 6 31 5 240 1 2 4 192 4 2 3 1 2 396 17 2 2 1 4 768 8 21 6 720 1 321 3 321 3 108 2 2 4 720 5 2 3 1 2 720 10 2 2 1 4 288 1 31 5 2 4 2,880 1 2 3 1 2 720 1 2 4 2 4 864 2 2 3 1 2 2,592 10 2 2 1 4 7,488 7 m z R z C z S  reg MC 8 431 2 4 21 6 17,280 2 2 3 1 2 2 3 1 2 3,744 15 2 2 1 4 3,456 7 42 2 3 2 1 2 3 2 1 2 8 1 32 2 1 16 1 321 3 48 1 2 4 192 2 2 3 1 2 336 4 2 2 1 4 576 3 32 2 1 32 2 1 72 8 321 3 240 10 31 5 720 2 2 4 384 4 2 3 1 2 1,104 23 2 2 1 4 2,880 15 21 6 5,760 2 321 3 321 3 360 4 2 4 1,728 6 2 3 1 2 2,448 17 2 2 1 4 1,728 3 31 5 2 4 5,760 1 2 3 1 2 2,880 1 2 4 2 4 1,296 4 2 3 1 2 5,184 11 2 2 1 4 19,584 12 21 6 69,120 3 1 8 241,920 1 2 3 1 2 2 3 1 2 10,368 24 2 2 1 4 15,552 15 21 6 8,640 1 2 2 1 4 2 2 1 4 3,456 2 3 2 2 3 2 2 3 2 2 4 1 3 2 1 2 8 1 32 2 1 48 4 321 3 144 4 31 5 480 1 2 4 192 3 2 3 1 2 720 11 2 2 1 4 2,640 11 21 6 10,080 3 1 8 40,320 1 3 2 1 2 3 2 1 2 16 1 32 2 1 104 7 321 3 240 5 31 5 480 1 2 4 480 4 2 3 1 2 1,032 14 2 2 1 4 1,920 7 21 6 1,440 1 32 2 1 32 2 1 396 29 321 3 1,020 43 31 5 2,640 6 2 4 1,440 15 2 3 1 2 4,008 84 2 2 1 4 9,792 51 21 6 18,720 7 321 3 321 3 1,440 12 31 5 720 1 2 4 4,032 14 2 3 1 2 6,336 44 2 2 1 4 5,184 9 m z R z C z S  reg MC 8 3 2 2 31 5 2 4 11,520 2 2 3 1 2 7,200 3 2 4 2 4 4,896 8 2 3 1 2 14,832 31 2 2 1 4 46,080 25 21 6 146,880 6 1 8 483,840 1 2 3 1 2 2 3 1 2 26,208 53 2 2 1 4 6,912 2 21 6 17,280 2 2 2 1 4 2 2 1 4 6,912 2 51 3 321 3 2 3 1 2 216 1 2 3 1 2 2 3 1 2 864 1 421 2 421 2 32 2 1 16 2 2 4 144 2 321 3 24 1 2 3 1 2 192 4 2 2 1 4 96 1 3 2 1 2 3 2 1 2 16 1 32 2 1 48 3 2 4 384 3 321 3 96 2 2 3 1 2 432 5 2 2 1 4 192 1 32 2 1 32 2 1 240 19 2 4 960 10 41 4 96 1 321 3 528 22 2 3 1 2 1,968 41 31 5 480 1 2 2 1 4 2,112 11 2 4 2 4 2,592 4 41 4 576 1 321 3 3,168 12 2 3 1 2 8,208 16 31 5 5,760 2 2 2 1 4 15,552 9 21 6 8,640 1 41 4 2 3 1 2 288 1 321 3 321 3 288 3 2 3 1 2 2,160 17 2 3 1 2 2 3 1 2 9,648 21 2 2 1 4 3,168 4 3 2 1 2 3 2 1 2 3 2 1 2 32 1 32 2 1 192 4 2 4 1,248 5 41 4 96 1 321 3 288 2 2 3 1 2 1,248 7 2 2 1 4 576 2 32 2 1 32 2 1 800 28 2 4 3,648 19 41 4 192 1 321 3 1,344 240 2 3 1 2 5,184 55 31 5 960 1 2 2 1 4 4,608 12 2 4 2 4 13,248 8 41 4 1,152 1 321 3 8,064 14 2 3 1 2 24,480 28 m z R z C z S  reg MC 8 3 2 1 2 2 4 31 5 11,520 1 2 2 1 4 38,016 14 21 6 17,280 1 41 4 2 3 1 2 576 1 321 3 321 3 576 3 2 3 1 2 4,176 15 2 3 1 2 2 3 1 2 19,296 23 2 2 1 4 5,184 5 32 2 1 32 2 1 32 2 1 2,768 69 2 4 9,504 59 41 4 720 6 321 3 5,328 117 2 3 1 2 18,144 206 31 5 8,640 11 2 2 1 4 26,016 77 21 6 15,840 5 2 4 2 4 27,072 16 41 4 2,304 2 321 3 22,176 77 2 3 1 2 62,784 110 31 5 48,960 9 2 2 1 4 130,176 57 21 6 207,360 7 41 4 321 3 432 2 2 3 1 2 2,880 5 321 3 321 3 4,078 31 2 3 1 2 19,512 137 2 2 1 4 4,896 9 2 3 1 2 2 3 1 2 72,576 133 31 5 8,640 4 2 2 1 4 47,232 42 2 4 2 4 2 4 67,824 8 41 4 5,184 2 321 3 69,120 14 2 3 1 2 177,120 25 31 5 172,800 3 2 2 1 4 475,200 20 21 6 1,296,000 5 1 8 3,628,800 2 41 4 41 4 576 1 321 3 3,456 2 2 3 1 2 12,096 3 2 2 1 4 3,456 1 321 3 321 3 27,216 22 2 3 1 2 90,720 54 31 5 8,640 1 2 2 1 4 58,752 10 2 3 1 2 2 3 1 2 263,952 53 31 5 86,400 3 2 2 1 4 302,400 30 21 6 129,600 2 2 2 1 4 2 2 1 4 51,840 4 41 4 2 3 1 2 2 3 1 2 4,320 2 321 3 321 3 2 3 1 2 4,752 10 2 3 1 2 2 3 1 2 36,288 24 2 3 1 2 2 3 1 2 2 3 1 2 167,184 27 2 2 1 4 33,696 7 Math. Meth. Appl. Si. 2017, 00 1{25 Copyright   2017 John Wiley & Sons, Ltd. 17 Prepared using mmaauth.ls Mathematial Methods in the Applied Sienes R. M. Falon, O. J. Falon, J. Nu~nez Shortly after, Lyakh [38℄ determined 21 ongurations with point rank 8, whih an be identied with the partial Latin squares 1 2 3 4 1 2 3 4 1234 3 4 1 2 1 2 3 4 2 1 4 3 1234 4 3 2 1 1 2 3 2 1 4 3 4 1 2 3 4 2 4 1 3 1234 1 3 4 2 F 1 F 2 F 3 F 4 F 5 F 6 F 7 2 4 4 1 2 3 1 3 2 4 4 1 2 3 3 1 2 4 1 3 4 3 2 1 2 4 3 1 4 3 2 1 3 2 4 1 3 2 4 1 4 1 3 2 2 3 4 1 3 4 2 1 2 3 4 1 F 8 F 9 F 10 F 11 F 12 F 13 F 14 132 321 2 1 423 321 1 4 243 2 1 3 1 4 234 132 4 1 3 4 2 123 4 1 342 2 1 3 4 1 432 321 4 1 F 15 F 16 F 17 F 18 F 19 F 20 F 21 They orrespond in Table 7to i. The two main lasses of type (4 2 ; 2 4 ; 2 4 ): F 3 and F 13 . ii. The four main lasses of type (42 2 ; 2 4 ; 2 4 ): F 2 , F 4 , F 6 and F 7 . iii. The main lass of type (3 2 2 ; 3 2 2 ; 3 2 2): F 15 . iv. The three main lasses of type (3 2 2 ; 3 2 2 ; 2 4 ): F 5 , F 12 and F 14 . v. The six main lasses of type (3 2 2 ; 2 4 ; 2 4 ): from F 16 to F 21 . vi. Five of the eight main lasses of type (2 4 ; 2 4 ; 2 4 ): F 1 , F 8 , F 9 , F 10 and F 11 . The next two main lasses of type (2 4 ; 2 4 ; 2 4 ) omplete the list of Lyakh. 1 2 2 1 3 4 4 3 1 2 3 4 4 2 3 1 F 22 F 23 The eighth main lass of type (2 4 ; 2 4 ; 2 4 ) is not related to a onguration beause there exist non-onneted points in the orresponding seminet (see Figure 8).  1 2 2 1 3 4 4 3 Figure 8. Seminet of point rank 8 that is not a onguration. 18 Copyright   2017 John Wiley & Sons, Ltd. Math. Meth. Appl. Si. 2017 ,001{25 Prepared using mmaauth.ls R. M. Falon, O. J. Falon, J. Nu~nez Mathematial Methods in the Applied Sienes 6. Binary onstraints related to the sets TS n and TCO n This setion deals with a series of binary onstraints that haraterize the sets of totally symmetri and totally onjugate orthogonal partial Latin squares of given order and weight. Hereafter, in order to avoid degeneray, partial Latin squares are assumed to have at least one entry in eah row, at least one entry in eah olumn, and at least one opy of eah symbol. From Theorem 2.1, the following system of onstraints must, therefore, hold.                                  x ijk x i 0 j k = 0 ; for all i ; i 0 ; j ; k  n suh that i 6 = i 0 ; x ijk x i j 0 k = 0 ; for all i ; j ; j 0 ; k  n suh that j 6 = j 0 ; x ijk x ijk 0 = 0 ; for all i ; j ; k ; k 0  n suh that k 6 = k 0 ; P j ;k 2 [ n ℄ x ijk  1 ; for all i 2 [ n ℄ ; P i ;k 2 [ n ℄ x ijk  1 ; for all j 2 [ n ℄ ; P i ;j 2 [ n ℄ x ijk  1 ; for all k 2 [ n ℄ ; x ijk 2 f 0 ; 1 g ; for all i ; j ; k  n : (2) Lemma 6.1 Let n and m be two positive integers suh that n  m  n 2 . a) If m > n , then every pair of orthogonal onjugates of a partial Latin square in the set TCO n ; m are distint. b) If j TCO n ; m j = 0 , then j TCO n ; m 0 j = 0 , for all m 0 2 f m + 1 ;:::;n 2 g . Pro of. Let us prove eah statement separately. a) Let P 2 R n;n ;n ; m and  ;  0 2 S 3 be suh that  6 =  0 and P  = P  0 . Sine m > n , there exists one symbol k 2 [ n ℄ and a distint pair of elements ( i 1 ; j 1 ) and, ( i 2 ; j 2 ) in [ n ℄  [ n ℄ suh that f ( i 1 ; j 1 ; k ) ; ( i 2 ; j 2 ; k ) g  E ( P  ) \ E ( P  0 ). As a onsequene, P  = P  0 is not orthogonal to itself. b) Otherwise, the partial Latin square that results after emptying any m 0  m lled ells of the partial Latin square in TCO n ; m 0 would be in TCO n ; m , whih is a ontradition. Lemma 6.1.a does not hold in general in ase of being m = n . Thus, for instane, the partial Latin square P 2 R 3 ; 3 ; 3;3 suh that E ( P ) = f (1 ; 1 ; 1) ; (2 ; 2 ; 2) ; (3 ; 3 ; 3) g is totally symmetri and orthogonal to itself. Based on (2), we establish in Setion 3 some equations to deal, respetively, with the sets TS n and TCO n . To this end, let us introdue the following notation x  i 1 i 2 i 3 := x i  (1) i  (2) i  (3) ; for all  2 S 3 and x i 1 i 2 i 3 2 f X g . Besides, we label the six permutations in S 3 as S 3 := f  1 = Id ;  2 = (12) ;  3 = (13) ;  4 = (23) ;  5 = (123) ;  6 = (132) g : Prop osition 6.2 Let n and m be two positive integers suh that n < m  n 2 . Then, a) The set TS n is identied with the set of zeros of (2) and x  s ijk = x ijk ; for all i ; j ; k 2 [ n ℄ and s 2 f 1 ; 2 ; 3 g : (3) b) The set TS n ; m is identied with the set of zeros of (2){(3) and X i ;j ;k 2 [ n ℄ x ijk = m : (4) Math. Meth. Appl. Si. 2017, 00 1{25 Copyright   2017 John Wiley & Sons, Ltd. 19 Prepared using mmaauth.ls Mathematial Methods in the Applied Sienes R. M. Falon, O. J. Falon, J. Nu~nez ) The set TCO n is identied with the set of zeros of (2) and x  s ijp x  s k l p x  t ijq x  t k l q = 0 ; for all i ; j ; k ; l ; p; q  n ; s ; t  3; suh that ( i ; j ) 6 = ( k ; l ) ; s  t : (5) d) The set TCO n ; m is identied with the set of zeros of (2), (4) and (5). Pro of. The result follows straightforwardly from the denitions exposed in Setion 2 one eah partial Latin square P = ( p i j ) 2 R r ;s ;n is identied with a zero ( x 111 ;:::; x r s n ) suh that x ijk = 1 if p i j = k and 0, otherwise. Thus, for instane, if we fous on the proof of statement (), then, given 1  s < t  3, the system of equations determined by (5) involves the   1 s - and   1 t -onjugates of P to be orthogonal. Besides, from Lemma 6.1.a, both onjugates are distint. Proposition 6.2 has been implemented in the CSP solver Minion [68℄ to obtain the numerial data exposed in Table 8. Further, Table 9indiates the run time that is required in our omputer system ( Intel Core i7-2600, with a 3.4 GHz proessor and 16 GB of RAM ) to determine one spei example in the sets TS n ; m and TCO n ; m . m j TS( n ; m ) j j TCO( n ; m ) j n n 3 4 5 6 3 4 3 1 36 4 6 1 216 576 5 6 12 1 12 45168 6 10 24 20 1 0 315048 7 12 64 80 30 0 391824 8 3 60 220 210 0 95028 9 3 100 380 680 0 2616 10 148 910 1980 0 11 72 1010 4380 0 12 90 1630 7660 0 13 72 2740 17820 0 14 36 2040 23370 0 15 16 2784 37476 0 16 16 3395 68850 0 17 2195 68190 18 2080 96660 19 2320 145560 20 900 122040 21 900 146040 22 480 196200 23 240 132480 24 30 148710 25 30 157320 26 101430 27 81540 28 86310 29 35820 30 33390 31 20340 32 11340 33 4560 34 3960 35 720 36 480 Total 41 711 24385 1755547 264 850260 Table 8. Distribution of the sets TS n ; m and TCO n ; m . 20 Copyright   2017 John Wiley & Sons, Ltd. Math. Meth. Appl. Si. 2017 ,001{25 Prepared using mmaauth.ls R. M. Falon, O. J. Falon, J. Nu~nez Mathematial Methods in the Applied Sienes Run time (seonds) Run time (seonds) n m TS n ; m TCO n ; m 5 5 < 1 22 10 < 1 3 6 6 < 1 8561 12 < 1 10 15 < 1 74 10 10 69 Out of memory 50 < 1 " 15 15 > 3 hours " 60 2 " 20 100 Out of memory " Table 9. Run times required to get exatly one totally symmetri or totally onjugate orthogonal partial Latin square of a given order and weight. 7. Lie partial quasigroup rings derived from the onjugate-extension of a partial Latin square The inlusion of new binary onstraints into (2){(5) enables us to determine families of partial Latin squares in the sets TS n and TCO n with possible appliations in distint elds. As an illustrative example, we onlude this paper by desribing in this setion a new family of Lie partial quasigroup rings related to a totally symmetri partial Latin square of order 3 n , whih is derived in turn from a given partial Latin square of order n . Reall that a Lie algebra is an anti-ommutative algebra A that holds the so-alled Jaobi identity J ( a; b ;  ) := ( ab )  + ( b  ) a + (  a ) b = 0 ; for all a; b ;  2 A: (6) Let P = ( p i j ) 2 R n;n ;n ; m . We dene the n  n arrays P 0 = ( p 0 i j ) and P 00 = ( p 00 i j ) suh that p 0 i j :=    p i j + n ; if p i j 2 [ n ℄ ; 0 ; otherwise : and p 00 i j :=    p i j + 2 n ; if p i j 2 [ n ℄ ; 0 ; otherwise : (7) Then, we dene the partial Latin square P = ( p i j ) 2 R 3 n; 3 n; 3 n ;6 m by means of nine n  n bloks as P :  0 P 00 P 0 (23) P 00 (12) 0 P (132) P 0 (123) P (13) 0 (8) where 0 denotes the n  n array with all its entries being zero. We all this new partial Latin square the onjugate-extension of P . Thus, for instane, Figure 9shows the onjugate-extension of the partial Latin square exposed in Figure 2. 7 8 4 5 9 5 7 6 7 1 8 9 1 2 7 3 4 6 1 3 5 1 5 2 Figure 9. Conjugate-extension of the partial Latin square P 2 R 3 ; 3 ; 3 of Figure 2. Math. Meth. Appl. Si. 2017, 00 1{25 Copyright   2017 John Wiley & Sons, Ltd. 21 Prepared using mmaauth.ls Mathematial Methods in the Applied Sienes R. M. Falon, O. J. Falon, J. Nu~nez Lemma 7.1 If P 2 R n;n ;n ; m , then P 2 TS 3 n ;6 m . Pro of. The result follows from the entry set E ( P ) one we keep in mind (7) and (8). Let A K ( P ) denote the partial quasigroup ring over a nite eld K of harateristi two that is related to P . Partiularly, we fous on the ase of being P 2 TS n . If this is the ase, then the denition (8) of the partial Latin square P results P  0 P 00 P 0 P 00 0 P P 0 P 0 (9) Theorem 7.2 Let K be a nite eld of harateristi two and let P 2 TS n be the multipliation table of a quasigroup ([ n ℄ ;  ) satisfying the left invertive law ( a  b )   = (   b )  a; for all a; b ;  2 [ n ℄ : (10) Then, the partial quasigroup ring A K ( P ) is a Lie algebra. Pro of. The symmetry of the partial Latin square P = ( p i j ), with p i i = 0, for all i  3 n , together with the fat of being K a nite eld of harateristi two, involves A K ( P ) to be anti-ommutative. Now, in order to prove that the Jaobi identity (6) holds, suppose f e 1 ;:::;e 3 n g to be the basis of A K ( P ), whih we partition into the three sets f e 1 ;:::;e n g , f e n +1 ;:::;e 2 n g and f e 2 n +1 ;:::;e 3 n g . Let S ( e i ) denote whih one of these three sets ontains eah basis vetor e i . From (9), we have that, if S ( e i ) = S ( e j ), then e i e j = 0. Besides, if S ( e i ) 6 = S ( e j ) and e i e j 6 = 0, then S ( e i ) 6 = S ( e i e j ) 6 = S ( e j ). As a onsequene, J ( e i ; e j ; e k ) = 0, for all i ; j ; k  3 n suh that the three sets S ( e i ), S ( e j ) and S ( e k ) either oinide or are pairwise distint. Then, from the symmetry of the Jaobi identity, it is enough to fous on the expression J ( e i ; e j ; e k ) in ase of being S ( e i ) = S ( e j ) 6 = S ( e k ). If this is the ase, e i e j = 0 and hene, J ( e i ; e j ; e k ) = ( e j e k ) e i + ( e k e i ) e j = e ( j  k )  i + e ( k  i )  j . The result follows from the symmetry of the partial Latin square P and the left invertive law. Every totally symmetri partial Latin square satisfying (10 ) onstitutes the multipliation table of a partial totally symmetri group. In order to ompute this kind of partial Latin squares, we inlude the following equations to (2){(4) x ijk x k l s x ljt ( x t i s  1) = 0 ; for all i ; j ; k ; l ; s ; t 2 [ n ℄ (11) X k  n x ijk  1 ! X k  n x ljk ! x ljt X k  n x t i k ! = 0 ; for all i ; j ; l ; t 2 [ n ℄ (12) x ijk X s  n x k l s  1 ! X s  n x ljs ! x ljt X s  n x t i s ! = 0 ; for all i ; j ; k ; l ; t 2 [ n ℄ (13) The implementation of these equations into our CSP solver determines, for instane, the pair of partial Latin squares exposed in Figure 10 , whih give rise in turn, aording to Theorem 7.2, to a pair of Lie partial quasigroup rings as we have previously desribed. 3 1 2 1 3 2 1 1 2 4 3 3 4 6 5 5 6 Figure 10. Totally symmetri partial Latin squares satisfying the left invertive law. 22 Copyright   2017 John Wiley & Sons, Ltd. Math. Meth. Appl. Si. 2017 ,001{25 Prepared using mmaauth.ls R. M. Falon, O. J. Falon, J. Nu~nez Mathematial Methods in the Applied Sienes 8. Conlusion and further studies This paper has dealt with the enumeration and lassiation of partial Latin retangles and seminets by means of omputational algebrai geometry. Both ombinatorial strutures have been identied with the points of aÆne varieties dened by zerodimensional radial ideals of polynomials. Their deompositions into nitely many disjoint subsets, eah of them being the zeros of a triangular system of polynomial equations, have emerged as a useful tehnique to determine, by means of the omputer algebra system Singular, the distribution of r  s partial Latin retangles based on [ n ℄ into isotopi and main lasses aording to their weight and types, for all r ; s ; n  6, and that of non-ompressible regular partial Latin squares of order n  8. The latter is equivalent to that of seminets with point rank up to eight and has enabled us to omplete a lassiation previously established by Lyakh [38℄. General formulas for the number of partial Latin squares of weight up to six and a ensus of all the seminets with at most six points have also been established. A onvenient generalization of the omputational method exposed in this paper to the theory of k -seminets and that of non-ompressible, regular and mutually regularly orthogonal partial Latin squares developed by Usan [12℄ is established as further work. We have also desribed a series of binary onstraints that enable us to determine the distribution of the sets TS n and TCO n of totally symmetri and totally onjugate partial Latin squares of order n , respetively, aording to their weights. By means of the CSP solver Minion, we have omputed the former, for all 2  n  6, and the latter, for all 2  n  4. A further study to improve the eÆieny of the proposed method is required to deal with higher orders. Besides, we have introdued the onjugate-extension of a given partial Latin square, whih gives rise to a totally symmetri partial Latin square. 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