Classifying partial Latin rectangles

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1 Raúl Falcón 1, Rebecca Stones 2 rafalgan@us.es 1 Department of Applied Mathematics I. University of Seville. 2 College of Computer and Control Engineering. Nankai University. Bergen. September 4, 2015.

2 Outline 1 Introduction. 2 Enumeration of PLR r,s,n. 3 Distribution into isomorphism and isotopism classes.

3 Outline 1 Introduction. 2 Enumeration of PLR r,s,n. 3 Distribution into isomorphism and isotopism classes.

4 Introduction Partial Latin rectangles. Algebraic Geometry.

5 Introduction Partial Latin rectangles. Algebraic Geometry.

6 The set PLR r,s,n. PLR r,s,n : Set of r s partial Latin rectangles based on [n] = {1,..., n}. 1 each cell is empty or contains one symbol of [n]. 2 each symbol occurs at most once in each row and each column. P PLR2,3,4 PLR2,3,5... PLR r,s,n;m : Subset of PLR r,s,n whose elements have m non-empty cells. P PLR 2,3,4;5 PLR 2,3,5;5... Entry set: {(row, column, symbol)} E(P) = {(1, 1, 1), (1, 3, 4), (2, 1, 4), (2, 2, 3), (2, 3, 2)}.

7 The set PLR r,s,n. PLR r,s,n : Set of r s partial Latin rectangles based on [n] = {1,..., n}. 1 each cell is empty or contains one symbol of [n]. 2 each symbol occurs at most once in each row and each column. P PLR2,3,4 PLR2,3,5... PLR r,s,n;m : Subset of PLR r,s,n whose elements have m non-empty cells. P PLR 2,3,4;5 PLR 2,3,5;5... Entry set: {(row, column, symbol)} E(P) = {(1, 1, 1), (1, 3, 4), (2, 1, 4), (2, 2, 3), (2, 3, 2)}.

8 The set PLR r,s,n. PLR r,s,n : Set of r s partial Latin rectangles based on [n] = {1,..., n}. 1 each cell is empty or contains one symbol of [n]. 2 each symbol occurs at most once in each row and each column. P PLR2,3,4 PLR2,3,5... PLR r,s,n;m : Subset of PLR r,s,n whose elements have m non-empty cells. P PLR 2,3,4;5 PLR 2,3,5;5... Entry set: {(row, column, symbol)} E(P) = {(1, 1, 1), (1, 3, 4), (2, 1, 4), (2, 2, 3), (2, 3, 2)}.

9 Symmetries on PLR r,s,n. S n : Symmetric group on [n]. S r S s S n : Isotopism group. P = Θ = ((1 2), (2 3), Id) S 2 S 3 S 3 P P Θ E(P) E(P Θ ) (1, 1, 1) (2, 1, 1) (1, 3, 3) (2, 2, 3) (2, 2, 3) (1, 3, 3) (2, 3, 2) (1, 2, 2) { Isotopism class : I n,p. P Θ = I n(p, Q) = {Θ S r S s S n P Θ = Q}

10 Symmetries on PLR r,s,n. S n : Symmetric group on [n]. S r S s S n : Isotopism group. P = Θ = ((1 2), (2 3), Id) S 2 S 3 S 3 P P Θ E(P) E(P Θ ) (1, 1, 1) (2, 1, 1) (1, 3, 3) (2, 2, 3) (2, 2, 3) (1, 3, 3) (2, 3, 2) (1, 2, 2) { Isotopism class : I n,p. P Θ = I n(p, Q) = {Θ S r S s S n P Θ = Q}

11 Symmetries on PLR r,s,n. S n : Symmetric group on [n]. S r S s S n : Isotopism group. P = Θ = ((1 2), (2 3), Id) S 2 S 3 S 3 E(P) E(P Θ ) (1, 1, 1) (2, 1, 1) (1, 3, 3) (2, 2, 3) (2, 2, 3) (1, 3, 3) (2, 3, 2) (1, 2, 2) P Θ = r = { s = n and α = β = γ Θ is an isomorphism ( ) Isomorphism class : I n,p. I n(p, Q) = {α S n P (α,α,α) = Q}

12 Symmetries on PLR r,s,n. Stabilizer groups: Autotopism group: A n (P) = I n (P, P). Automorphism group: A n (P) = I n (P, P). PLR Θ = {P PLR r,s,n Θ A n (P)}. PLR Θ;m = {P PLR r,s,n;m Θ A n (P)}. Lemma a) A n(p) = A n(q), Q I n(p). b) I n,p = r! s! n! A n(p).

13 Symmetries on PLR r,s,n. Stabilizer groups: Autotopism group: A n (P) = I n (P, P). Automorphism group: A n (P) = I n (P, P). PLR Θ = {P PLR r,s,n Θ A n (P)}. PLR Θ;m = {P PLR r,s,n;m Θ A n (P)}. Lemma a) A n(p) = A n(q), Q I n(p). b) I n,p = r! s! n! A n(p).

14 Enumerating and classiying PLRs. The enumeration and classification problems of PLRs are open. Case r = s = n and m = n 2 : Latin squares. n 11: [McKay, Wanless; 2005][Hulpke, Kaski, Östergård; 2011]. Case r s = n and m = rn: Latin rectangles. n 11: [Stones; 2010]. General case r s n and m rs: Partial Latin rectangles. r, s, n 4: Enumeration [Falcón; 2012; 2015]. r, s, n 7: Enumeration and classification [Falcón, Stones; Under preparation]. 1 Inclusion-exclusion method. 2 Chromatic polynomial method. 3 Sade s method. 4 Algebraic Geometry method.

15 Enumerating and classiying PLRs. The enumeration and classification problems of PLRs are open. Case r = s = n and m = n 2 : Latin squares. n 11: [McKay, Wanless; 2005][Hulpke, Kaski, Östergård; 2011]. Case r s = n and m = rn: Latin rectangles. n 11: [Stones; 2010]. General case r s n and m rs: Partial Latin rectangles. r, s, n 4: Enumeration [Falcón; 2012; 2015]. r, s, n 7: Enumeration and classification [Falcón, Stones; Under preparation]. 1 Inclusion-exclusion method. 2 Chromatic polynomial method. 3 Sade s method. 4 Algebraic Geometry method.

16 Enumerating and classiying PLRs. The enumeration and classification problems of PLRs are open. Case r = s = n and m = n 2 : Latin squares. n 11: [McKay, Wanless; 2005][Hulpke, Kaski, Östergård; 2011]. Case r s = n and m = rn: Latin rectangles. n 11: [Stones; 2010]. General case r s n and m rs: Partial Latin rectangles. r, s, n 4: Enumeration [Falcón; 2012; 2015]. r, s, n 7: Enumeration and classification [Falcón, Stones; Under preparation]. 1 Inclusion-exclusion method. 2 Chromatic polynomial method. 3 Sade s method. 4 Algebraic Geometry method.

17 Enumerating and classiying PLRs. Rr,s,n:m r.s.n m , ,208 1,278 3,552 9,696 25, ,332 6,504 3,078 13,716 58, , ,008 9,792 3,834 29, ,864 1,437, ,104 2,412 36, ,064 5,728, , , ,200 15,326, , ,688 27,534, , ,584 32,971, ,744 25,941, ,368 13,153, ,215, , , , Total ,601 28,353 11, ,425 2,423, ,545,137 Proposition (Falcón, 2015) a) PLR r,s,n:0 = 1. b) PLR r,s,n:1 = rsn. c) PLR r,s,n:2 = 1 2rsn (rsn r s n + 2).

18 Enumerating and classiying PLRs. Rr,s,n:m r.s.n m , ,208 1,278 3,552 9,696 25, ,332 6,504 3,078 13,716 58, , ,008 9,792 3,834 29, ,864 1,437, ,104 2,412 36, ,064 5,728, , , ,200 15,326, , ,688 27,534, , ,584 32,971, ,744 25,941, ,368 13,153, ,215, , , , Total ,601 28,353 11, ,425 2,423, ,545,137 Proposition (Falcón, 2015) a) PLR r,s,n:0 = 1. b) PLR r,s,n:1 = rsn. c) PLR r,s,n:2 = 1 2rsn (rsn r s n + 2).

19 Introduction Partial Latin rectangles. Algebraic Geometry.

20 Algebraic Geometry F [x] : Ring of polynomials in x = {x 1,..., x n } over a field F. I : An ideal in F [x]. Standard grading: F [x] = 0 d F [x] d. x a = x a x an n is identified with a = (a 1,..., a n ) N n. A monomial term order < is a total order on N n, where a) The zero vector is the unique minimal element. b) a < b a + c < b + c, for all a, b, c N n. Example: Graded lexicographic term order: x 1 x 2 x 2 3 < lex x 2 1 x 2 x 3 < lex x 2 1 x 2 x 2 3. The largest monomial of a polynomial is its initial monomial. x 1 x 2 x x 2 1 x 2 x 3 + x 2 1x 2 x 2 3. A monomial is standard if it is not contained in the ideal generated by all the initial monomials.

21 Algebraic Geometry F [x] : Ring of polynomials in x = {x 1,..., x n } over a field F. I : An ideal in F [x]. Standard grading: F [x] = 0 d F [x] d. x a = x a x an n is identified with a = (a 1,..., a n ) N n. A monomial term order < is a total order on N n, where a) The zero vector is the unique minimal element. b) a < b a + c < b + c, for all a, b, c N n. Example: Graded lexicographic term order: x 1 x 2 x 2 3 < lex x 2 1 x 2 x 3 < lex x 2 1 x 2 x 2 3. The largest monomial of a polynomial is its initial monomial. x 1 x 2 x x 2 1 x 2 x 3 + x 2 1x 2 x 2 3. A monomial is standard if it is not contained in the ideal generated by all the initial monomials.

22 Algebraic Geometry F [x] : Ring of polynomials in x = {x 1,..., x n } over a field F. I : An ideal in F [x]. Standard grading: F [x] = 0 d F [x] d. x a = x a x an n is identified with a = (a 1,..., a n ) N n. A monomial term order < is a total order on N n, where a) The zero vector is the unique minimal element. b) a < b a + c < b + c, for all a, b, c N n. Example: Graded lexicographic term order: x 1 x 2 x 2 3 < lex x 2 1 x 2 x 3 < lex x 2 1 x 2 x 2 3. The largest monomial of a polynomial is its initial monomial. x 1 x 2 x x 2 1 x 2 x 3 + x 2 1x 2 x 2 3. A monomial is standard if it is not contained in the ideal generated by all the initial monomials.

23 Algebraic Geometry F [x] : Ring of polynomials in x = {x 1,..., x n } over a field F. I : An ideal in F [x]. Standard grading: F [x] = 0 d F [x] d. x a = x a x an n is identified with a = (a 1,..., a n ) N n. A monomial term order < is a total order on N n, where a) The zero vector is the unique minimal element. b) a < b a + c < b + c, for all a, b, c N n. Example: Graded lexicographic term order: x 1 x 2 x 2 3 < lex x 2 1 x 2 x 3 < lex x 2 1 x 2 x 2 3. The largest monomial of a polynomial is its initial monomial. x 1 x 2 x x 2 1 x 2 x 3 + x 2 1x 2 x 2 3. A monomial is standard if it is not contained in the ideal generated by all the initial monomials.

24 Algebraic Geometry F [x] : Ring of polynomials in x = {x 1,..., x n } over a field F. I : An ideal in F [x]. Standard grading: F [x] = 0 d F [x] d. x a = x a x an n is identified with a = (a 1,..., a n ) N n. A monomial term order < is a total order on N n, where a) The zero vector is the unique minimal element. b) a < b a + c < b + c, for all a, b, c N n. Example: Graded lexicographic term order: x 1 x 2 x 2 3 < lex x 2 1 x 2 x 3 < lex x 2 1 x 2 x 2 3. The largest monomial of a polynomial is its initial monomial. x 1 x 2 x x 2 1 x 2 x 3 + x 2 1x 2 x 2 3. A monomial is standard if it is not contained in the ideal generated by all the initial monomials.

25 Algebraic Geometry F [x] : Ring of polynomials in x = {x 1,..., x n } over a field F. I : An ideal in F [x]. Standard grading: F [x] = 0 d F [x] d. x a = x a x an n is identified with a = (a 1,..., a n ) N n. A monomial term order < is a total order on N n, where a) The zero vector is the unique minimal element. b) a < b a + c < b + c, for all a, b, c N n. Example: Graded lexicographic term order: x 1 x 2 x 2 3 < lex x 2 1 x 2 x 3 < lex x 2 1 x 2 x 2 3. The largest monomial of a polynomial is its initial monomial. x 1 x 2 x x 2 1 x 2 x 3 + x 2 1x 2 x 2 3. A monomial is standard if it is not contained in the ideal generated by all the initial monomials.

26 Algebraic Geometry Algebraic set V(I ) = {a F n : f (a) = 0 for all f I }. If I is zero-dimensional and radical, then V(I ) = dim F F [x]/i = 0 d HF F [x]/i (d), where HF is the Hilbert function HF (d) = dim F (F [x] d /I d ). It coincides with the number of standard monomials in I of degree d with respect to any term order. [Moeller, 1993] [Hillebrand, 1999] [Dickenstein, 2012].

27 Algebraic Geometry Algebraic set V(I ) = {a F n : f (a) = 0 for all f I }. If I is zero-dimensional and radical, then V(I ) = dim F F [x]/i = 0 d HF F [x]/i (d), where HF is the Hilbert function HF (d) = dim F (F [x] d /I d ). It coincides with the number of standard monomials in I of degree d with respect to any term order. [Moeller, 1993] [Hillebrand, 1999] [Dickenstein, 2012].

28 Algebraic Geometry F 2 [x]: Ring of polynomials in x = {x 1,..., x rsn } over the finite field F 2. Theorem (Falcón, 2015) The set PLR r,s,n is identified with the set of zeros in F 2 [x] of the ideal I r,s,n generated by Then, x ijk (x ijk 1), i [r], j [s], k [n], x ijk x ijk, i [r], j [s], k, k [n], k < k, x ijk x ij k, i [r], j, j [s], k [n], j < j, x ijk x i jk, i, i [r], j [s], k [n], i < i. a) PLR r,s,n = dim F2 (F 2 [x]/i r,s,n ) b) R r,s,n:m = HF F2[x]/I r,s,n (m), for all m 0. (1) P = x111 1, x113 1, x223 1, x232 1, x112, x113,...

29 Algebraic Geometry PLRr,s,n n r s ,045 1,961 3, ,546 4,051 9,276 19, ,327 37,633 93, , , ,441, ,447 2,605 4, ,601 12,781 37,273 93, , , , ,181 2,599,185 7,985, ,502,171 10,291,951 54,730, ,605, ,694, ,744,637 5,376,213, ,256,288,925 92,842,518, ,219,832,671, , , ,366 4,193,269 17,464,756 60,983, ,423,521 33,199, ,651,473 2,263,521,961 12,703,477, ,442,316 15,916,515, ,463,431,546 1,854,072,020, ,905,708,889 11,785,736,969,413 * ,545,137 4,146,833,121 87,136,329,169 1,258,840,124,753 * 5 313,185,347,701 * * * *Excessive cost of computation for a computer system i7-2600, 3.4 GHz. Max. time of computation: 4,180 seconds (PLR 2,9,13 ).

30 Algebraic Geometry PLRr,s,n n r s ,021 1,354 1,753 2, ,509 8,501 12,585 18,001 25, ,046 63, , , , , , ,597 1,442,173 2,456, ,376 2,501,801 5,470,158 11,109,337 21,204, ,596,553 12,975,561 32,989,969 76,751, ,625, ,572,114 58,941, ,721, ,630,861 1,163,391, ,662, ,073, ,128,454 7,307,593, ,405,357,682 12,470,162,233 40,864,292, ,334,454, ,976,401, ,324,308, ,985 10,411 15,137 21,325 29, , ,161 1,458,733 2,482,801 4,050, ,582,613 52,585, ,667, ,278, ,597, ,742,231 2,756,029,891 7,846,852,421 20,336,594,221 48,689,098, ,175,825, ,690,051, ,999,864,193 1,398,636,508,477 4,141,988,637, ,377,377,305 3,836,955,565,101 18,712,512,041,917 78,819,926,380, ,220,109,353, ,372,136,371,721 99,423,049,782, ,303,240,153,313 3,595,671,023,722,081 17,076,864,830,330, ,156,152,706,483 2,000,246,352,476,311 17,908,872,286,407, ,297,226,011,020, ,986,548,443,056, ,296,831,902,738, ,203,526,838,449,441 * * 11 * * * ,952, ,317,981 1,231,810,504 2,803,520,281 5,970,344, ,737,345, ,769,858, ,139,572,873 2,487,656,927,521 7,030,865,002, ,451,823,665,776 * * * * *Excessive cost of computation for a computer system i7-2600, 3.4 GHz. Max. time of computation: 4,180 seconds (PLR 2,9,13 ).

31 Outline 1 Introduction. 2 Enumeration of PLR r,s,n. 3 Distribution into isomorphism and isotopism classes.

32 An algebraic alternative to compute PLR r,s,n The new approach is based on the symmetry of those polynomials in (1) related to each row of a partial Latin rectangle. x ijk (x ijk 1), i [r], j [s], k [n], x ijk x ijk, i [r], j [s], k, k [n], k < k, x ijk x ij k, i [r], j, j [s], k [n], j < j, x ijk x i jk, i, i [r], j [s], k [n], i < i.

33 An algebraic alternative to compute PLR r,s,n 1 For each i [r], let I (i) r,s,n = x ijk x ij k, x ijk x ijk : j, j [s], k, k [n], j < j, k < k 2 Let {J1,1,..., J 1,t } be a finite set of t subideals of I r,s,n (1) generated by triangular systems of polynomial equations and whose affine varieties constitute a partition of V (I r,s,n). (1) [Moeller, 1993] [Hillebrand, 1999] 3 For i > 1, let Ji,l be the subideal of I (i) r,s,n whose generators coincide with those of J 1,l after replacing each variable x 1jk by x ijk. 4 For each tuple (t1,..., t r ) [t] r, let K t1,...,t r = J 1,t J r,tr + x ijk x i jk : i, i [r], j [s], k [n], i < i

34 An algebraic alternative to compute PLR r,s,n 1 For each i [r], let I (i) r,s,n = x ijk x ij k, x ijk x ijk : j, j [s], k, k [n], j < j, k < k 2 Let {J1,1,..., J 1,t } be a finite set of t subideals of I r,s,n (1) generated by triangular systems of polynomial equations and whose affine varieties constitute a partition of V (I r,s,n). (1) [Moeller, 1993] [Hillebrand, 1999] 3 For i > 1, let Ji,l be the subideal of I (i) r,s,n whose generators coincide with those of J 1,l after replacing each variable x 1jk by x ijk. 4 For each tuple (t1,..., t r ) [t] r, let K t1,...,t r = J 1,t J r,tr + x ijk x i jk : i, i [r], j [s], k [n], i < i

35 An algebraic alternative to compute PLR r,s,n 1 For each i [r], let I (i) r,s,n = x ijk x ij k, x ijk x ijk : j, j [s], k, k [n], j < j, k < k 2 Let {J1,1,..., J 1,t } be a finite set of t subideals of I r,s,n (1) generated by triangular systems of polynomial equations and whose affine varieties constitute a partition of V (I r,s,n). (1) [Moeller, 1993] [Hillebrand, 1999] 3 For i > 1, let Ji,l be the subideal of I (i) r,s,n whose generators coincide with those of J 1,l after replacing each variable x 1jk by x ijk. 4 For each tuple (t1,..., t r ) [t] r, let K t1,...,t r = J 1,t J r,tr + x ijk x i jk : i, i [r], j [s], k [n], i < i

36 An algebraic alternative to compute PLR r,s,n 1 For each i [r], let I (i) r,s,n = x ijk x ij k, x ijk x ijk : j, j [s], k, k [n], j < j, k < k 2 Let {J1,1,..., J 1,t } be a finite set of t subideals of I r,s,n (1) generated by triangular systems of polynomial equations and whose affine varieties constitute a partition of V (I r,s,n). (1) [Moeller, 1993] [Hillebrand, 1999] 3 For i > 1, let Ji,l be the subideal of I (i) r,s,n whose generators coincide with those of J 1,l after replacing each variable x 1jk by x ijk. 4 For each tuple (t1,..., t r ) [t] r, let K t1,...,t r = J 1,t J r,tr + x ijk x i jk : i, i [r], j [s], k [n], i < i

37 An algebraic alternative to compute PLR r,s,n The triangularity of the systems involve each subideal J i,tj to have at least one generator of the form x ij k or x ij k 1. The number m t1,...,t r of generators x ij k 1 in K t1,...,t r constitutes the minimum number of cells that are necessarily non-empty in any partial Latin rectangle identified with a point of the affine variety V (K t1,...,t r ). Proposition Let m be a non-negative integer. Then PLR r,s,n;m = HF F2[x]/I r,s,n (m) = HF (m m ). F2[x]/Kt1,...,tr t1,...,tr (t 1,...,t r ) [t] r m t1,...,tr m Advantages: Less storage memory. Parallel computation.

38 An algebraic alternative to compute PLR r,s,n The triangularity of the systems involve each subideal J i,tj to have at least one generator of the form x ij k or x ij k 1. The number m t1,...,t r of generators x ij k 1 in K t1,...,t r constitutes the minimum number of cells that are necessarily non-empty in any partial Latin rectangle identified with a point of the affine variety V (K t1,...,t r ). Proposition Let m be a non-negative integer. Then PLR r,s,n;m = HF F2[x]/I r,s,n (m) = HF (m m ). F2[x]/Kt1,...,tr t1,...,tr (t 1,...,t r ) [t] r m t1,...,tr m Advantages: Less storage memory. Parallel computation.

39 An algebraic alternative to compute PLR r,s,n The triangularity of the systems involve each subideal J i,tj to have at least one generator of the form x ij k or x ij k 1. The number m t1,...,t r of generators x ij k 1 in K t1,...,t r constitutes the minimum number of cells that are necessarily non-empty in any partial Latin rectangle identified with a point of the affine variety V (K t1,...,t r ). Proposition Let m be a non-negative integer. Then PLR r,s,n;m = HF F2[x]/I r,s,n (m) = HF (m m ). F2[x]/Kt1,...,tr t1,...,tr (t 1,...,t r ) [t] r m t1,...,tr m Advantages: Less storage memory. Parallel computation.

40 An algebraic alternative to compute PLR r,s,n The triangularity of the systems involve each subideal J i,tj to have at least one generator of the form x ij k or x ij k 1. The number m t1,...,t r of generators x ij k 1 in K t1,...,t r constitutes the minimum number of cells that are necessarily non-empty in any partial Latin rectangle identified with a point of the affine variety V (K t1,...,t r ). Proposition Let m be a non-negative integer. Then PLR r,s,n;m = HF F2[x]/I r,s,n (m) = HF (m m ). F2[x]/Kt1,...,tr t1,...,tr (t 1,...,t r ) [t] r m t1,...,tr m Advantages: Less storage memory. Parallel computation.

41 Example: PLR 3,3,3 The ideal I (1) 3,3,3 can be decomposed into the next six disjoint subideals J 1,1 = I (1) 3,3,3 + x 111, x 121, x 131. J 1,2 = I (1) 3,3,3 + x 111, x 121, x 131 1, x 132, x 133. J 1,3 = I (1) 3,3,3 + x 111, x 121 1, x 122, x 123, x 131. J 1,4 = I (1) 3,3,3 + x 111 1, x 112, x 113, x 121, x 122, x 131, x 132. J 1,5 = I (1) 3,3,3 + x 111 1, x 112, x 113, x 121, x 122, x 131, x 132 1, x 133. J 1,6 = I (1) 3,3,3 + x 111 1, x 112, x 113, x 121, x 122 1, x 123, x 131, x 132. For each triple (t 1, t 2, t 3 ) [6] 3, let K t1,t 2,t 3 = J 1,t1 + J 2,t2 + J 3,t3 + x ijk x i jk : i, i, j, k [3], i < i

42 Example: PLR 3,3,3 The ideal I (1) 3,3,3 can be decomposed into the next six disjoint subideals J 1,1 = I (1) 3,3,3 + x 111, x 121, x 131. J 1,2 = I (1) 3,3,3 + x 111, x 121, x 131 1, x 132, x 133. J 1,3 = I (1) 3,3,3 + x 111, x 121 1, x 122, x 123, x 131. J 1,4 = I (1) 3,3,3 + x 111 1, x 112, x 113, x 121, x 122, x 131, x 132. J 1,5 = I (1) 3,3,3 + x 111 1, x 112, x 113, x 121, x 122, x 131, x 132 1, x 133. J 1,6 = I (1) 3,3,3 + x 111 1, x 112, x 113, x 121, x 122 1, x 123, x 131, x 132. For each triple (t 1, t 2, t 3 ) [6] 3, let K t1,t 2,t 3 = J 1,t1 + J 2,t2 + J 3,t3 + x ijk x i jk : i, i, j, k [3], i < i

43 Example: PLR 3,3,3 HF F2[x]/Kt 1,t 2,t 3 (m) t1.t2.t3 m Example: PLR 3,3,3:2 = (t 1,t 2,t 3) [6] 3 m t1,t 2,t 3 2 HF F2[x]/K t1,t 2,t 3 (2 m t1,t 2,t 3 ) = HF F2[x]/K 1,1,1 (2) + 3 HF F2[x]/K 1,1,2 (1) + 3 HF F2[x]/K 1,1,3 (1)+ 3 HF F2[x]/K 1,1,4 (1) + 3 HF F2[x]/K 1,1,5 (0) + 3 HF F2[x]/K 1,1,6 (0)+ 6 HF F2[x]/K 1,2,3 (0) + 6 HF F2[x]/K 1,2,4 (0) + 6 HF F2[x]/K 1,3,4 (0) = = 270. K 1,2,3 1 1 m 1,2,3 = 2.

44 Example: PLR 3,3,3 HF F2[x]/Kt 1,t 2,t 3 (m) t1.t2.t3 m Example: PLR 3,3,3:2 = (t 1,t 2,t 3) [6] 3 m t1,t 2,t 3 2 HF F2[x]/K t1,t 2,t 3 (2 m t1,t 2,t 3 ) = HF F2[x]/K 1,1,1 (2) + 3 HF F2[x]/K 1,1,2 (1) + 3 HF F2[x]/K 1,1,3 (1)+ 3 HF F2[x]/K 1,1,4 (1) + 3 HF F2[x]/K 1,1,5 (0) + 3 HF F2[x]/K 1,1,6 (0)+ 6 HF F2[x]/K 1,2,3 (0) + 6 HF F2[x]/K 1,2,4 (0) + 6 HF F2[x]/K 1,3,4 (0) = = 270. K 1,2,3 1 1 m 1,2,3 = 2.

45 Enumeration of PLR r,s,n;m Rr,s,n:m r.s.n m , ,208 1,278 3,552 9,696 25, ,332 6,504 3,078 13,716 58, , ,008 9,792 3,834 29, ,864 1,437, ,104 2,412 36, ,064 5,728, , , ,200 15,326, , ,688 27,534, , ,584 32,971, ,744 25,941, ,368 13,153, ,215, , , , Total ,601 28,353 11, ,425 2,423, ,545,137

46 Enumeration of PLR r,s,n;m Rr,s,5:m r.s m , ,500 2,400 2,760 4,400 7, ,680 4,800 10,400 7,590 20,520 43,200 54, , , ,140 20,040 61,400 40, , , ,200 1,881,600 5,159, ,680 45, , , ,360 3,594,960 5,641,920 21,612,480 80,602, ,920 54, , ,680 3,018,000 17,930,400 32,423, ,546, ,160, , , ,840 6,605,280 60,912, ,248,960 1,045,147,200 7,845,192, , , ,520 9,224, ,826, ,731,360 4,530,640,800 50,648,616, ,600 27,480 7,983, ,307, ,440,320 14,444,083, ,687,408, ,280 4,063, ,419,040 1,004,380,800 33,852,910, ,069,668, ,100, ,010, ,238,720 58,065,734,400 2,741,210,616, ,960 59,047, ,722,880 72,278,294,400 6,104,066,712, ,284, ,580,800 64,484,985,600 10,385,299,320, ,512,000 63,884,160 40,544,726,400 13,420,351,008, ,240 9,216,000 17,571,260,160 13,065,814,483, ,400 5,099,169,600 9,486,099,648, ,107,200 5,073,056,640, ,288,000 1,970,474,400, ,681, ,608,096, , ,330,054, ,667,552, ,388,160, ,008, ,032, ,280 Total , , ,661 1,502, ,366 33,199, ,442,316 4,146,833, ,185,347,701 64,170,718,937,006

47 Enumeration of PLR r,s,n;m Rr,s,6:m r.s.6 m ,470 2,160 1,188 2,196 3,510 5, ,200 2, ,120 8,880 19,200 35,400 13,896 37,344 78, , ,800 5, ,990 48, , ,650 94, ,580 1,115,100 2,547, ,320 15, , ,240 2,168, ,340 2,676,240 10,667,160 31,419, , ,760 1,899,600 8,546, ,380 12,082,680 70,540, ,470, ,480 3,139,200 21,211,200 1,375,920 36,270, ,808,000 1,727,352, ,160 2,881,800 32,189,400 1,038,960 71,633,160 1,064,140,200 7,893,282, ,303,200 28,267, ,760 90,585,600 2,422,568,400 26,212,965, ,480 13,063,680 69,603,840 3,803,369,040 62,938,898, ,669,760 29,255,040 4,021,099, ,045,861, ,800 5,112,000 2,756,361, ,246,779, ,152,144, ,367,120, ,828,800 58,252,478, ,791,040 19,683,613, ,828,798, ,652, ,321,600 Total ,045 4,051 13,327 1,447 37, ,181 10,291, ,694,843 4,193, ,651,473 15,916,515, ,905,708,889

48 Enumeration of PLR r,s,n;m Rr,s,6:m r.s.6 m ,032 6,420 9,360 10,200 14,850 21, , , , , ,400 1,330, ,384,24 4,245,120 9,527,220 11,532,600 25,631,100 56,614, ,476,480 62,189, ,310, ,154, ,260,640 1,771,796, ,148, ,375,600 2,434,907,520 3,352,566,000 12,019,602, ,57,620, ,176,640 5,189,068,800 25,231,996,800 37,450,656, ,585,456, ,416,600, ,599,625,880 30,548,079, ,165,742, ,946,451,000 1,991,858,418,000 11,852,197,317, ,281,623, ,625,603,200 1,226,542,944,000 2,171,483,394,000 18,056,836,776, ,993,809,528, ,618,868, ,097,055,680 5,834,154,055,680 11,456,637,616, ,095,655,863,200 1,406,144,941,776, ,771,527,680 1,152,338,169,600 21,579,415,960,320 47,586,889,008, ,225,199,808,000 11,344,829,123,448, ,451,767,040 2,190,542,918,400 62,007,749,812, ,763,852,264,000 3,616,441,279,056,000 75,444,662,621,250, ,491,532,800 3,099,028,723, ,935,650,124, ,342,211,504,000 13,801,803,749,280, ,809,990,051,328, ,635,215,360 3,221,159,616, ,112,048,230, ,559,781,792,000 42,582,496,312,944,000 1,888,965,825,155,136, ,337,610,240 2,415,807,221, ,313,104,578,560 1,281,622,863,052, ,042,151,250,892,000 7,129,083,890,074,291, ,116,800 1,274,532,969, ,524,671,011,840 1,569,898,647,504, ,529,994,957,440,000 22,290,972,757,613,899, ,792,486, ,831,824,435,200 1,478,352,018,528, ,378,166,715,776,000 57,672,207,579,205,440, ,134,464, ,967,540,608,000 1,058,153,580,288, ,045,603,416,704, ,205,370,805,154,944, ,604,889,600 44,468,899,430, ,490,862,592, ,874,461,303,296, ,689,524,093,737,792, ,854,080 11,483,903,278, ,899,017,011, ,217,521,389,081, ,570,613,181,156,803, ,942,917,304,320 63,429,754,752, ,606,324,462,848, ,084,100,503,749,939, ,499,481,600 12,467,229,696, ,166,400,104,064, ,915,364,298,700,288, ,670,220,800 1,610,606,592,000 36,987,139,952,640, ,409,503,369,430,016, ,046, ,628,032,000 9,853,601,458,752, ,607,699,757,168,896, ,356,218,880 1,909,729,461,012,480 82,187,524,303,374,458, ,267,391,462,400 31,703,766,748,202,926, ,634,533,888,000 9,523,824,649,261,056, ,496,724,480,000 2,204,514,949,427,712, ,752,384, ,140,940,150,784, ,851,200 51,905,194,846,617, ,196,712,196,505, ,383,137,792, ,862,379,520, ,655,040, ,262,643, ,851,200 Total 87,136,329,169 14,554,896,138,901 1,474,670,894,380,885 7,687,297,409,633,551 2,322,817,844,850,427, ,7032,853,070,203,981,647

49 Outline 1 Introduction. 2 Enumeration of PLR r,s,n. 3 Distribution into isomorphism and isotopism classes.

50 Isotopism and isomorphism clases We focus on the distribution of PLR r,s,n into isomorphism and isotopism classes, that is, on the sets I r,s,n and I r,s,n. Lemma Let r, s and n be three positive integers. If rs n, then I r,s,n = I r,s,rs. If s n, then I 1,s,n = s + 1. Let PLR Θ denote the set of r s partial Latin rectangles based on [n] that have an isotopism Θ S r S s S n in its autotopism group. Lemma Let Θ 1 and Θ 2 be two conjugate isotopisms in S r S s S n. Then, PLR Θ1 = PLR Θ2. The set of isotopism classes of PLR Θ1 coincides with that of PLR Θ2.

51 Isotopism and isomorphism clases We focus on the distribution of PLR r,s,n into isomorphism and isotopism classes, that is, on the sets I r,s,n and I r,s,n. Lemma Let r, s and n be three positive integers. If rs n, then I r,s,n = I r,s,rs. If s n, then I 1,s,n = s + 1. Let PLR Θ denote the set of r s partial Latin rectangles based on [n] that have an isotopism Θ S r S s S n in its autotopism group. Lemma Let Θ 1 and Θ 2 be two conjugate isotopisms in S r S s S n. Then, PLR Θ1 = PLR Θ2. The set of isotopism classes of PLR Θ1 coincides with that of PLR Θ2.

52 Isotopism and isomorphism clases To be conjugate is an equivalence relation among isotopisms. Each conjugacy class is characterized by the common cycle structure of its elements. The cycle structure of a permutation π in the symmetric group S m is defined as the expression z π = m d π m... 1 d π 1, where d π i is the number of cycles of length i in the unique cycle decomposition of the permutation π. The cycle structure of an isotopism (α, β, γ) S r S s S n is the triple z Θ = (z α, z β, z γ ). Θ = ((1234), (12)(3)(45), (12)(345)(6)) S 4 S 5 S 6 z Θ = (4, 2 2 1, 321).

53 Isotopism and isomorphism clases CS m : The set of cycle structures of the symmetric group S m. Let z CS m. Let di z of cycle structure z. denote the value of d π i for all permutation π S m Lemma A triple z = (z 1, z 2, z 3 ) CS r CS s CS n is the cycle structure of an isotopism of a non-empty r s partial Latin rectangle based on [n] if and only if there exists a triple (i, j, k) [r] [s] [n] such that lcm(i, j) = lcm(i, k) = lcm(j, k) = lcm(i, j, k) and d z1 i d z2 j d z3 k > 0.

54 Isotopism and isomorphism clases CS m : The set of cycle structures of the symmetric group S m. Let z CS m. Let di z of cycle structure z. denote the value of d π i for all permutation π S m Lemma A triple z = (z 1, z 2, z 3 ) CS r CS s CS n is the cycle structure of an isotopism of a non-empty r s partial Latin rectangle based on [n] if and only if there exists a triple (i, j, k) [r] [s] [n] such that lcm(i, j) = lcm(i, k) = lcm(j, k) = lcm(i, j, k) and d z1 i d z2 j d z3 k > 0.

55 Isotopism and isomorphism clases (z 1, z 2, z 3 ) = PLR Θ for any isotopism Θ of cycle structure (z 1, z 2, z 3 ) CS r CS s CS n. Since the isotopism and the isomorphism groups are finite groups that acts on R r,s,n and R n,n,n, respectively, Burnside s lemma and the number of permutations with a given cycle structure involve: I r,s,n = α S r β S s γ S n (z α, z β, z γ ) r!s!n! = z 1 CS r z 2 CS s z 3 CS n I n = π S n (z π, z π, z π ) n! i [r] j [s] k [n] d z1 i (z 1, z 2, z 3 )!d z2 j!d z3 k! i d z 1 i j d z 2 j k d z 3 k = (z, z, z) z CS n i [n] d i z!i d. i z.

56 Isotopism and isomorphism clases (z 1, z 2, z 3 ) = PLR Θ for any isotopism Θ of cycle structure (z 1, z 2, z 3 ) CS r CS s CS n. Since the isotopism and the isomorphism groups are finite groups that acts on R r,s,n and R n,n,n, respectively, Burnside s lemma and the number of permutations with a given cycle structure involve: I r,s,n = α S r β S s γ S n (z α, z β, z γ ) r!s!n! = z 1 CS r z 2 CS s z 3 CS n I n = π S n (z π, z π, z π ) n! i [r] j [s] k [n] d z1 i (z 1, z 2, z 3 )!d z2 j!d z3 k! i d z 1 i j d z 2 j k d z 3 k = (z, z, z) z CS n i [n] d i z!i d. i z.

57 Isotopism and isomorphism clases Theorem Let Θ = (α, β, γ) be an isotopism of PLR r,s,n. The set PLR Θ is identified with the affine variety defined by the ideal in F 2 [x 111,..., x rsn ] I Θ = I r,s,n x ijk x α(i)β(j)γ(k) : i [r], j [s], k [n]. Further, (z α, z β, z γ ) = PLR Θ = dim F2 (F 2 [x 111,..., x rsn ]/I Θ ).

58 Isotopism and isomorphism clases n I n r s n I r,s,n r s n I r,s,n

59 References Adams, P., R. Bean and A. Khodkar, A census of critical sets in the Latin squares of order at most six, Ars Comb. 68 (2003), Falcón, R. M., Cycle structures of autotopisms of the Latin squares of order up to 11, Ars Comb. 103 (2012), Falcón, R. M., The set of autotopisms of partial Latin squares, Discrete Math. 313 (2013), Falcón, R. M., Enumeration and classification of self-orthogonal partial Latin rectangles by using the polynomial method, European J. Combin. 48 (2015), R. M. Falcón and R. J. Stones. Enumerating partial Latin rectangles (2015). In preparation. Decker, W., G. M. Greuel, G. Pfister and H. Schönemann, Singular A computer algebra system for polynomial computations (2015), URL: Hillebrand, D., Triangulierung nulldimensionaler ideale - implementierung und vergleich zweier algorithmen, Master s thesis, Universitaet Dortmund, Fachbereich Mathematik, Hulpke, A., P. Kaski and P. R. J. Östergård, The number of Latin squares of order 11, Math. Comp. 80 (2011), McKay, B. D. and I. M. Wanless, On the number of Latin squares, Ann. Comb. 9 (2005), McKay, B. D., A. Meynert and W. Myrvold, Small Latin Squares, Quasigroups and Loops, J. Combin. Des. 15 (2007), Möller, H. M., On decomposing systems of polynomial equations with finitely many solutions, Appl. Algebra Engrg. Comm. Comput. 4 (1993), Stones, D. S., P. Vojtĕchovský and I. M. Wanless, Cycle structure of autotopisms of quasigroups and Latin squares, J. Combin. Des. 20 (2012),

60 THANK YOU!!! Raúl Falcón 1, Rebecca Stones 2 rafalgan@us.es 1 Department of Applied Mathematics I. University of Seville. 2 College of Computer and Control Engineering. Nankai University. Bergen. September 4, 2015.

Invariants for efficiently computing the autotopism group of a partial Latin rectangle

Invariants for efficiently computing the autotopism group of a partial Latin rectangle Daniel Kotlar Tel-Hai College, Israel joint work with Eiran Danan, Raúl M. Falcón, Trent G. Marbach, Rebecca J. Stones