What Do Lattice Paths Have To Do With Matrices, And What Is Beyond Both?

Transcription

1 What Do Lattice Paths Have To Do With Matrices, And What Is Beyond Both? Joseph E. Bonin The George Washington University These slides are available at blogs.gwu.edu/jbonin Some ideas in this talk were developed jointly with Anna de Mier and Marc Noy at Universitat Politècnica de Catalunya, Barcelona

2 The story line: start with some simple objects, namely, lattice paths, connect these objects with some things you may know, explore the structure of these objects, generalize the objects, connect the generalizations with things you may know, look at the generalizations from many perspectives, re-examine our original objects from these perspectives, see that this all falls under a much broader theory, and offer glimpses of that broader theory.

3 Lattice Paths Lattice paths are sequences of east and north steps of unit length, starting at (0, 0). (6, 4) (0, 0)

4 Lattice Paths Lattice paths are sequences of east and north steps of unit length, starting at (0, 0). (6, 4) (0, 0)

5 Lattice Paths Lattice paths are sequences of east and north steps of unit length, starting at (0, 0). (6, 4) (0, 0)

6 Lattice Paths Lattice paths are sequences of east and north steps of unit length, starting at (0, 0). (6, 4) (0, 0)

7 Lattice Paths Lattice paths are sequences of east and north steps of unit length, starting at (0, 0). (6, 4) (0, 0)

8 Lattice Paths Lattice paths are sequences of east and north steps of unit length, starting at (0, 0). (6, 4) (0, 0)

9 Lattice Paths Lattice paths are sequences of east and north steps of unit length, starting at (0, 0). (6, 4) (0, 0)

10 Lattice Paths Lattice paths are sequences of east and north steps of unit length, starting at (0, 0). (6, 4) (0, 0)

11 Lattice Paths Lattice paths are sequences of east and north steps of unit length, starting at (0, 0). (6, 4) (0, 0)

12 Lattice Paths Lattice paths are sequences of east and north steps of unit length, starting at (0, 0). (6, 4) (0, 0)

13 Lattice Paths Lattice paths are sequences of east and north steps of unit length, starting at (0, 0). (6, 4) (0, 0)

14 Sets of Lattice Paths Fix lattice paths P and Q from (0, 0) to (m, r) with P never going above Q. Let P be the set of lattice paths from (0, 0) to (m, r) that stay in the region that P and Q bound. Q P

15 Sets of Lattice Paths Fix lattice paths P and Q from (0, 0) to (m, r) with P never going above Q. Let P be the set of lattice paths from (0, 0) to (m, r) that stay in the region that P and Q bound. Q P What is P?

16 A Simple Example (m, r) For P = E m N r and Q = N r E m, we get ( ) m + r P =. r (0, 0)

17 Finding the Number of Paths Recursively P can be computed recursively. a a + b b a a 1. a 1 2 a

18 Finding the Number of Paths Recursively P can be computed recursively. a a + b b a a 1 3. a a

19 Finding the Number of Paths Recursively P can be computed recursively. a a + b a b a a a

20 Finding the Number of Paths Recursively P can be computed recursively. a a + b a b a a a

21 Finding the Number of Paths Recursively P can be computed recursively. a a + b a b a a a

22 Finding the Number of Paths Recursively P can be computed recursively. a a a + b a b a a On a grid, we see the identity Pascal s triangle ( ) n = k ( n 1 k ) ( ) n 1 and k 1.

23 Paths as Transversals of Set Systems Each path in P is determined by where its north steps occur. The lattice path with north steps at steps 1, 3, 7,

24 Paths as Transversals of Set Systems Each path in P is determined by where its north steps occur. The lattice path with north steps at steps 1, 3, 7, Label each north step by its position in each path it is in.

25 Paths as Transversals of Set Systems Each path in P is determined by where its north steps occur. The lattice path with north steps at steps 1, 3, 7, Label each north step by its position in each path it is in. N 4 = {6, 7, 8, 9, 10} N 3 = {4, 5, 6, 7} N 2 = {2, 3, 4, 5, 6} N 1 = {1, 2, 3, 4} For 1 i r, let N i be the set of labels on the north steps in row i of the diagram (from the bottom).

26 Set Systems, Transversals, and Partial Transversals A set system is a (finite) multiset A = (A j : j J) of subsets of a (finite) set S. Example: S = {a, b, c, d, e, f, g, h} and A = (A, B, C, D) with A = {a, b, e, f, h}, B = {b, c, g}, C = {d, e, g, h}, D = {d, f, h} A B C D a b c d e f g h

27 Set Systems, Transversals, and Partial Transversals A set system is a multiset A = (A j : j J) of subsets of a set S. A = {a, b, e, f, h}, B = {b, c, g}, C = {d, e, g, h}, D = {d, f, h} A transversal of A is a set {x j : j J} of J distinct elements with x j A j for all j J. Some transversals: {a, b, e, h}, {e, f, g, h}. A B C D a b c d e f g h

28 Set Systems, Transversals, and Partial Transversals Note: a transversal is just a set whose elements can be matched with the sets in A; the matchings are not part of the transversal. A = {a, b, e, f, h}, B = {b, c, g}, C = {d, e, g, h}, D = {d, f, h} A = {a, b, e, f, h}, B = {b, c, g}, C = {d, e, g, h}, D = {d, f, h} A B C D a b c d e f g h

29 Set Systems, Transversals, and Partial Transversals A = {a, b, e, f, h}, B = {b, c, g}, C = {d, e, g, h}, D = {d, f, h} A transversal of A is a set {x j : j J} of J distinct elements with x j A j for all j J. A partial transversal of A is a transversal of some subsystem (A k : k K) with K J. Some partial transversals: {a, b, e}, {c},.

30 Set Systems, Transversals, and Partial Transversals A = {a, b, e, f, h}, B = {b, c, g}, C = {d, e, g, h}, D = {d, f, h} A transversal of A is a set {x j : j J} of J distinct elements with x j A j for all j J. A partial transversal of A is a transversal of some subsystem (A k : k K) with K J. Some partial transversals: {a, b, e}, {c},. Some sets that are not partial transversals: {a, b, c}, {a, b, e, g}.

31 A Matrix Perspective on Set Systems and Partial Transversals A = {a, b, e, f, h}, B = {b, c, g}, C = {d, e, g, h}, D = {d, f, h} A B C D a b c d e f g h

32 A Matrix Perspective on Set Systems and Partial Transversals A = {a, b, e, f, h}, B = {b, c, g}, C = {d, e, g, h}, D = {d, f, h} A B C D a b c d e f g h x A,a x A,b 0 0 x A,e x A,f 0 x A,h 0 x B,b x B,c x B,g x C,d x C,e 0 x C,g x C,h x D,d 0 x D,f 0 x D,h Replace s with distinct variables.

33 A Matrix Perspective on Set Systems and Partial Transversals A = {a, b, e, f, h}, B = {b, c, g}, C = {d, e, g, h}, D = {d, f, h} A B C D a b c d e f g h x A,a x A,b 0 0 x A,e x A,f 0 x A,h 0 x B,b x B,c x B,g x C,d x C,e 0 x C,g x C,h x D,d 0 x D,f 0 x D,h Replace s with distinct variables. The determinant of the red submatrix is x A,b x B,g x C,e x D,h x A,e x B,b x C,g x D,h.

34 The Permutation Expansion of a Determinant The 2 by 2 case: ( ) c11 c det 12 = c c 21 c 11 c 22 c 12 c 21, 22 c 11 c 12 c 13 For 3 by 3 matrices, det c 21 c 22 c 23 is c 31 c 32 c 33 ( ) c22 c c 11 det 23 c c 32 c 12 det 33 which is ( c21 c 23 c 31 c 33 ( ) ( ) ) + c 13 det ( c21 c 22 c 31 c 32 c 11 c 22 c 33 c 11 c 23 c 32 c 12 c 21 c 33 +c 12 c 23 c 31 +c 13 c 21 c 32 c 13 c 22 c 31. ),

35 The Permutation Expansion of a Determinant For an n n matrix C = (c i,j ), det(c) = σ sgn(σ) c 1,σ(1) c 2,σ(2) c n,σ(n) where we sum over all permutations σ : {1,..., n} {1,..., n} and sgn(σ) is 1 (resp. 1) if σ results from an even (resp. odd) number of transpositions.

36 The Permutation Expansion of a Determinant For an n n matrix C = (c i,j ), det(c) = σ sgn(σ) c 1,σ(1) c 2,σ(2) c n,σ(n) where we sum over all permutations σ : {1,..., n} {1,..., n} and sgn(σ) is 1 (resp. 1) if σ results from an even (resp. odd) number of transpositions. From this definition, it looks like a determinant requires roughly n! operations to compute, but, by using other properties of determinants, they can be found with roughly n 3 operations.

37 The Determinants of Interest for Partial Transversals x A,b x A,e 0 x A,h det x B,b 0 x B,g 0 0 x C,e x C,g x C,h = x A,bx B,g x C,e x D,h x A,e x B,b x C,g x D,h x D,h The determinant of a square submatrix with rows indexed by R {A, B, C, D} and columns indexed by C {a, b,..., h} is the sum, over all matchings φ : C R, of the product of the variables expressing the matching, ± i C x φ(i),i. (Non-matchings (where i φ(i) for some i C) have a zero in the corresponding product.)

38 The Determinants of Interest for Partial Transversals x A,b x A,e 0 x A,h det x B,b 0 x B,g 0 0 x C,e x C,g x C,h = x A,bx B,g x C,e x D,h x A,e x B,b x C,g x D,h x D,h The determinant of a square submatrix with rows indexed by R {A, B, C, D} and columns indexed by C {a, b,..., h} is the sum, over all matchings φ : C R, of the product of the variables expressing the matching, ± i C x φ(i),i. (Non-matchings (where i φ(i) for some i C) have a zero in the corresponding product.) The nonzero entries are distinct variables, so no products cancel.

39 The Determinants of Interest for Partial Transversals x A,b x A,e 0 x A,h det x B,b 0 x B,g 0 0 x C,e x C,g x C,h = x A,bx B,g x C,e x D,h x A,e x B,b x C,g x D,h x D,h The determinant of a square submatrix with rows indexed by R {A, B, C, D} and columns indexed by C {a, b,..., h} is the sum, over all matchings φ : C R, of the product of the variables expressing the matching, ± i C x φ(i),i. (Non-matchings (where i φ(i) for some i C) have a zero in the corresponding product.) The nonzero entries are distinct variables, so no products cancel. Thus, four columns correspond to a transversal if and only if the determinant of the corresponding 4 4 submatrix is nonzero.

40 Getting Matrices over the Reals Now replace the variables by certain real numbers. A B C D a b c d e f g h r A,a r A,b 0 0 r A,e r A,f 0 r A,h 0 r B,b r B,c r B,g r C,d r C,e 0 r C,g r C,h r D,d 0 r D,f 0 r D,h The information about the partial transversals is preserved if the determinants of the square submatrices that were nonzero before remain nonzero. It is possible to do this with non-negative real numbers and with each column sum being 1.

41 Convex Hulls and the Simplex Vectors with non-negative real entries that add to 1 are the points in the convex hull of the standard basis vectors. (a, b) = a(1, 0) + b(0, 1), a 0, b 0, a + b = 1 (1, 0) (a, b) (0, 1) The 1-simplex, a line segment. x + y = 1

42 Convex Hulls and the Simplex Vectors with non-negative real entries that add to 1 are the points in the convex hull of the standard basis vectors. (0, 0, 1) (a, b, c) (1, 0, 0) (0, 1, 0) The 2-simplex, a triangle. In the plane x + y + z = 1 in R 3. a 0, b 0, c 0, a + b + c = 1 (a, b, c) = a(1, 0, 0) + b(0, 1, 0) + c(0, 0, 1) ( b = a(1, 0, 0) + (b + c) (0, 1, 0) + c ) (0, 0, 1) b + c b + c

43 Convex Hulls and the Simplex Vectors with non-negative real entries that add to 1 are the points in the convex hull of the standard basis vectors. (0, 0, 1) (a, b, c) (1, 0, 0) (0, 1, 0) The 2-simplex, a triangle. In the plane x + y + z = 1 in R 3. a 0, b 0, c 0, a + b + c = 1 (a, b, c) = a(1, 0, 0) + b(0, 1, 0) + c(0, 0, 1) ( b = a(1, 0, 0) + (b + c) (0, 1, 0) + c ) (0, 0, 1) b + c b + c The 3-simplex is a tetrahedron with vertices (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1) in the hyperplane x + y + z + w = 1 of R 4, etc.

44 A Geometric Perspective on Set Systems and Partial Transversals Each column can be assigned non-negative entries that sum to 1, so each column can be seen as a point in the convex hull of r affinely independent points, v 1, v 2,..., v r, in R r 1. e 3 A B C D a b c d e f g h e 4 d f h e 1 e a g b c e 2 Thus, all information about partial transversals is captured by the placement of the elements on the faces of a simplex in R r 1. The actual coordinates of the elements are of no importance; what matters is which elements are in which faces.

45 What do we get if we focus only on the independence or dependence of sets of points, not on coordinates?

46 What do we get if we focus only on the independence or dependence of sets of points, not on coordinates? A MATROID!

47 Matroids A matroid M is a finite set E and a set I of subsets of E, the independent sets, such that (i) I, (ii) if A I and B A, then B I, and (iii) if A, B I and B < A, then B {x} I for some x A B. (Whitney, 1935) Example. Let the elements of E label the columns of a matrix. A subset of E is independent iff the corresponding columns are distinct and linearly independent.

48 Matroids A matroid M is a finite set E and a set I of subsets of E, the independent sets, such that (i) I, (ii) if A I and B A, then B I, and (iii) if A, B I and B < A, then B {x} I for some x A B. (Whitney, 1935) Example. Let the elements of E label the columns of a matrix. A subset of E is independent iff the corresponding columns are distinct and linearly independent. The bases are maximal independent sets. Note that all bases have the same size.

49 Transversal Matroids Theorem (Edmonds and Fulkerson, 1965) The partial transversals of a set system A are the independent sets of a matroid.

50 Transversal Matroids Theorem (Edmonds and Fulkerson, 1965) The partial transversals of a set system A are the independent sets of a matroid. A is a presentation of this transversal matroid.

51 Transversal Matroids Theorem (Edmonds and Fulkerson, 1965) The partial transversals of a set system A are the independent sets of a matroid. A is a presentation of this transversal matroid. Theorem (Brylawski, 1975) A matroid is transversal if and only if it has a geometric realization on a simplex with all dependence arising from the structure of the simplex. Arising from the structure of the simplex means, for example, if {a, b, c} is dependent but all of its 2-subsets are independent, then a, b, c must be on an edge of the simplex. d f h e a g b c

52 Lattice Path Matroids Let P and Q be lattice paths from (0, 0) to (m, r) with P never going above Q. Let P and N 1, N 2,..., N r be as before. N 4 = {6, 7, 8, 9, 10} N 3 = {4, 5, 6, 7} N 2 = {2, 3, 4, 5, 6} N 1 = {1, 2, 3, 4} Q P The matroid M[P, Q] is the transversal matroid on {1, 2,..., m + r} with presentation (N 1, N 2,..., N r ). A lattice path matroid is a matroid that is isomorphic to (i.e., up to relabeling, is) M[P, Q] for some such P and Q.

53 An Example in Detail 6 7 N 3 = {6, 7} N 2 = {3, 4, 5, 6} N 1 = {1, 2, 3, 4} {6, 7} 7 N 3 N 2 N {1, 2, 3, 4} {3, 4, 5, 6}

54 Bases and Paths The bases of a matroid are its maximal independent sets. Theorem R {i the i-th step of R is north} is a bijection between P and the set of bases of M[P, Q]

55 Bases and Paths The bases of a matroid are its maximal independent sets. Theorem R {i the i-th step of R is north} is a bijection between P and the set of bases of M[P, Q]

56 Bases and Paths The bases of a matroid are its maximal independent sets. Theorem R {i the i-th step of R is north} is a bijection between P and the set of bases of M[P, Q]

57 Duality

58 Duality The dual M of M has {E B B is a basis of M} as its set of bases. Theorem The class of lattice path matroids is closed under duality.

59 Duality The class of transversal matroids is not closed under duality. a a b b c c a b c a b c

60 Deletion Given a matroid M on E and x E not in all bases of M, the deletion M\x is the matroid on E x whose bases are all bases B of M with x B. {6, 7} 7 {6, 7} {1, 2, 3, 4} {3, 4, 5, 6} {1, 2, 3} {3, 5, 6}

61 Contraction Contraction is the dual operation: M/x = (M \x). If x is in at least one basis of M, then the bases of M/x are the sets B x as B ranges over the bases of M that contain x. {6, 7} {1, 2, 3, 4} {3, 4, 5, 6} 6 1, 2, 3, {1, 2, 3, 5, 6} {6, 7}

62 Minors If x is either in no bases, or in all bases, then we set M\x = M/x. Minors are obtained by iterating deletion and contraction. Theorem The class of lattice path matroids is closed under minors. This paves the way for a characterization the class of lattice path matroids by determining the minor-minimal obstructions to membership in the class

63 A Counterpart of Hamiltonian Circuits Theorem If the paths P and Q meet only at the ends, (0, 0) and (m, r), then there is a set of r + 1 elements in M[P, Q] all of whose r-element subsets are bases...

64 A Counterpart of Hamiltonian Circuits Theorem If the paths P and Q meet only at the ends, (0, 0) and (m, r), then there is a set of r + 1 elements in M[P, Q] all of whose r-element subsets are bases...

65 A Counterpart of Hamiltonian Circuits Theorem If the paths P and Q meet only at the ends, (0, 0) and (m, r), then there is a set of r + 1 elements in M[P, Q] all of whose r-element subsets are bases...

66 A Counterpart of Hamiltonian Circuits Theorem If the paths P and Q meet only at the ends, (0, 0) and (m, r), then there is a set of r + 1 elements in M[P, Q] all of whose r-element subsets are bases...

67 A Change of Focus: A Brief Overview of Some of the Many Research Directions in Matroid Theory

68 Representability A matroid is representable over a field F if it is isomorphic to (i.e., a relabeling of) the matroid on the columns of some matrix over F given by linear independence.

69 Representability A matroid is representable over a field F if it is isomorphic to (i.e., a relabeling of) the matroid on the columns of some matrix over F given by linear independence. Theorem Welsh, 1970) (Piff and A transversal matroid is representable over every sufficiently large field, in particular, over every infinite field. matroids transversal matroids representable lattice path matroids over matroids representable over R duals of transversal some field

70 Representability Not all matroids are representable over fields. (Mac Lane, 1936) Projective geometry is the geometry of subspaces of vector spaces. To get a non-representable matroid, take the configuration in a theorem of projective geometry and alter it to violate the theorem but to give a valid matroid configuration.

71 A Non-Representable Matroid Related to Desargues Theorem Desargues Theorem In a planar configuration, two triangles are perspective from a point if and only if they are perspective from a line. Desargues configuration A non-representable matroid Perspective from a point: the lines through corresp. vertices intersect in a point. Perspective from a line: the points of intersection of corresp. sides are on a line.

72 A Non-Representable Matroid Related to Pappus Theorem Pappus Theorem If the vertices of a hexagon lie alternately on two coplanar lines, then the three points in which pairs of opposite sides intersect are collinear Pappus configuration A non-representable matroid

73 Representability is Rare Theorem (Nelson, 2018) Almost all matroids are not representable over any field. He proved an upper bound of 2 n3 /4 on the number of representable matroids on n elements. A lower bound of 2 ( n n/2)/n n log n on the number of matroids on n elements was proven by Knuth (1974).

74 The Problem of Characterizing Representable Matroids If M is representable over F, then so are its minors. Thus, one way to describe which matroids are representable over F is to find the minor-minimal matroids that are not. These minimal obstructions are the excluded minors.

75 The Problem of Characterizing Representable Matroids If M is representable over F, then so are its minors. Thus, one way to describe which matroids are representable over F is to find the minor-minimal matroids that are not. These minimal obstructions are the excluded minors. The field F 2 of size two Theorem (Tutte, 1958) A matroid is representable over F 2 (binary) iff it does not have U 2,4 as a minor. U 2,4

76 More Representability Results Theorem (Tutte, 1958) A matroid is representable over all fields iff it has none of U 2,4, F 7, and F 7 as minors. F 7

77 More Representability Results Theorem (Tutte, 1958) A matroid is representable over all fields iff it has none of U 2,4, F 7, and F 7 as minors. F 7 Theorem (Reid, 1972, unpub.; Seymour, 1979; Bixby, 1979) The excluded minors for representability over F 3 (ternary matroids) are U 2,5, U 3,5, F 7, and F

78 A Well-Known Excluded-Minor Result in Graph Theory Geelen, Gerards, and Kapoor (2000) proved the excluded minor characterization of representability over F 4. There are seven excluded minors x 1 + x x 1 + x x x x x 1 + x x 1 + x x x 1 + x x 1 + x x 0 x 1 + x x x 1 x

79 A Well-Known Excluded-Minor Result in Graph Theory Geelen, Gerards, and Kapoor (2000) proved the excluded minor characterization of representability over F 4. There are seven excluded minors x 1 + x x 1 + x x x x x 1 + x x 1 + x x x 1 + x x 1 + x x 0 x 1 + x x x 1 x Currently, over 500 excluded minors for representability over F 5 are known.

80 A Well-Known Excluded-Minor Result in Graph Theory Kuratowski s Theorem A graph is planar (i.e., it can be drawn in the plane with no edges that cross) if and only if neither K 5 nor K 3,3 is a minor. K 5 K 3,3

81 A Well-Known Excluded-Minor Result in Graph Theory Kuratowski s Theorem A graph is planar (i.e., it can be drawn in the plane with no edges that cross) if and only if neither K 5 nor K 3,3 is a minor. K 5 K 3,3 K 5 K 5 K 3,3

82 A Well-Known Excluded-Minor Result in Graph Theory Kuratowski s Theorem A graph is planar (i.e., it can be drawn in the plane with no edges that cross) if and only if neither K 5 nor K 3,3 is a minor. K 5 K 3,3 The Petersen graph.

83 A Well-Known Excluded-Minor Result in Graph Theory Kuratowski s Theorem A graph is planar (i.e., it can be drawn in the plane with no edges that cross) if and only if neither K 5 nor K 3,3 is a minor. K 5 K 3,3 The Petersen graph. graph.

84 A Well-Known Excluded-Minor Result in Graph Theory Kuratowski s Theorem A graph is planar (i.e., it can be drawn in the plane with no edges that cross) if and only if neither K 5 nor K 3,3 is a minor. K 5 K 3,3 The Petersen graph. graph. graph.

85 Several Major Theorems about Minors The Graph Minors Theorem (Robertson and Seymour) Any minor-closed class of graphs has only finitely many excluded minors. Theorem (Geelen, Gerards, Whittle, announced 2013) For a finite field F, there are only finitely many excluded minors for representability over F. (Rota s conjecture (1971)) The Matroid Minors Theorem announced, 2013) (Geelen, Gerards, Whittle, If F is a finite field, then any proper minor-closed class of F-representable matroids has only finitely many excluded minors.

86 More Directions in Matroid Theory Extremal matroid theory: How many elements can a matroid have if it satisfies certain conditions? In general, find inequalities relating various parameters of a matroid. Constructions: New operations continue to be discovered. Some old operations are far from fully understood. Algebraic matroids. Unimodality. Generalizations, matroids with extra structure, and variations: greedoids, jump systems, Coxeter matroids, flag matroids, polymatroids, bimatroids, delta matroids, oriented matroids, antimatroids, and much, much more.

87 Thank you for listening.