An Introduction to Transversal Matroids

Transcription

1 An Introduction to Transversal Matroids Joseph E Bonin The George Washington University These slides and an accompanying expository paper (in essence, notes for this talk, and more) are available at jbonin/

2 Part I Several Perspectives on Transversal Matroids

3 Set Systems 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} Each set system gives rise to a bipartite graph, and vice versa A B C D a b c d e f g h

4 Transversals A set system is a multiset A = (A j : j J) of subsets of a set S A transversal of A is a subset T of S for which there is a bijection (matching) ψ : T J with t A ψ(t) for all t T system of distinct representatives Some transversals: {a,b,e,h}, {e,f,g,h} A B C D a b c d e f g h 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} Caution: a transversal is just a set whose elements can be matched with the sets in A; the matchings are not part of the transversal

5 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 subset T of S for which there is a bijection ψ : T J with t A ψ(t) for all t T A partial transversal of A is a transversal of a subsystem, (A k : k K) with K J Some partial transversals: {a, b, e}, {c},, {a, b, d, f } Some sets that are not partial transversals: {a,b,c}, {a,b,e,g}

6 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 s are algebraically independent

7 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 b e g h x A,b x A,e 0 x A,h x B,b 0 x B,g 0 0 x C,e x C,g x C,h x D,h s are algebraically independent The determinant of this 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

8 The Permutation Expansion of Determinants Reflects Matchings 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 and columns indexed by C is the sum, over all matchings φ : C R, of the product arising from the matching, ie, ± x φ(i),i i C (Non-matchings (where i φ(i) for some i C) have a zero in the corresponding product and so make no contribution) The nonzero entries are algebraically independent, so there is no cancellation, so partial transversals correspond to linearly independent sets of columns

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

10 Transversal Matroids Theorem The partial transversals of a set system A are the independent sets of a matroid (Edmonds and Fulkerson, 1965) A is a presentation of this transversal matroid, M[A] Theorem A transversal matroid is representable over all sufficiently large fields, in particular, over all infinite fields (Piff and Welsh, 1970) Likewise, we can always find a matrix representation with nonnegative real entries

11 Results Easily Seen from the Set System Perspective Theorem The class of transversal matroids is closed under direct sums Proof by group discussion

12 Results Easily Seen from the Set System Perspective Theorem The class of transversal matroids is closed under direct sums Proof by group discussion Theorem If M = M[A 1,,A r ], then M X = M[A 1 X,,A r X] for any X E(M) Thus, the class of transversal matroids is closed under restriction

13 A Result Easily Seen from the Matrix Perspective Theorem Any transversal matroid M of rank r has a presentation by r sets Indeed, if M = M[A 1,,A k ] and if some basis {b 1,,b r } is a transversal of (A 1,,A r ), then M = M[A 1,,A r ]

14 A Result Easily Seen from the Matrix Perspective Theorem Any transversal matroid M of rank r has a presentation by r sets Indeed, if M = M[A 1,,A k ] and if some basis {b 1,,b r } is a transversal of (A 1,,A r ), then M = M[A 1,,A r ] Proof b 1,,b r, A 1 X Z A r Y S Observe: the first r rows span the row space

15 A Result Easily Seen from the Matrix Perspective Theorem Any transversal matroid M of rank r has a presentation by r sets Indeed, if M = M[A 1,,A k ] and if some basis {b 1,,b r } is a transversal of (A 1,,A r ), then M = M[A 1,,A r ] Proof b 1,,b r, A 1 X Y 0 0 A r Observe: the first r rows span the row space

16 A Corollary If (A 1,,A r ) and (A 1,,A k ), with r(m) = r < k, are both presentations of M, then all elements in A r+1 A k are coloops Corollary If M has no coloops, then all presentations of M have r(m) nonempty sets

17 Background Cyclic Flats A set X in a matroid M is cyclic if X is a union of circuits; equivalently, M X has no coloops Let Z(M) be the set of cyclic flats of M {a, b, c, d, e, f, g, h} c a d b g e h f {a, b, c, d} {e, f, g, h} e a f b g c h d

18 Cyclic Flats and Their Ranks Determine a Matroid A matroid is determined by its cyclic flats and their ranks (Brylawski, 1975) {a, b, c, d, e, f, g, h} c a d b M g e h f {a, b, c, d} {e, f, g, h} 3 2 e a g f c b N h d

19 Cyclic Flats and Their Ranks Determine a Matroid A matroid is determined by its cyclic flats and their ranks (Brylawski, 1975) {a, b, c, d, e, f, g, h} c a d b M g e h f {a, b, c, d} {e, f, g, h} 3 2 e a g f c b N h d Group discussion Find some other ranks that we can assign to these sets to get matroids

20 Cyclic Flats in Transversal Matroids If M = M[A 1,,A r ], then M X = M[A 1 X,,A r X] for any X E(M) If M has no coloops, then all presentations of M have r(m) nonempty sets Corollary If F is a cyclic flat of M = M[A 1,,A r ], then there are exactly r(f) integers i with F A i

21 A Geometric Representation on a Simplex In a representing matrix with nonnegative real entries, scale each nonzero column so the column sum is 1 A B C D a b c d e f g h 1 p 0 0 r s 0 u 0 1 p t q 1 r 0 1 t v q 0 1 s 0 1 u v Nonzero columns in such a matrix are points in the convex hull of the standard basis vectors This convex hull is a simplex c b a f e 2 e e 4 1 h g e d e 3

22 A Geometric Perspective on Transversal Matroids Theorem A matroid is transversal iff it has a geometric realization on a simplex in which each cyclic flat of rank k is the set of points in some face of the simplex spanned by k vertices (Brylawski, 1975) c e 2 b a f e e 4 1 h g e d e 3

23 Some Results Easily Seen from the Geometric Perspective A rank-r transversal matroid has at most ( r k) cyclic flats of rank k

24 Some Results Easily Seen from the Geometric Perspective A rank-r transversal matroid has at most ( r k) cyclic flats of rank k The class of transversal matroids is not closed under duality a b b a c d c d e f e f

25 Some Results Easily Seen from the Geometric Perspective A rank-r transversal matroid has at most ( r k) cyclic flats of rank k The class of transversal matroids is not closed under duality a b b a c d c d e f e f Nor under contraction x M M/x ***

26 Summary of Part 1 set systems bipartite graphs matrix representations partial transversals sets meeting matchings independent sets transversal matroids and presentations r(m) sets suffice in a presentation; without coloops, all presentations have r(m) nonempty sets geometric representations; rank-k cyclic flats are placed in k-vertex faces of the simplex closure under direct sums and deletions failure of closure under duality and contractions

27 Problem Session 1 Consider ({1,2,,8}, A) and ({1,2,,8}, A ) with A = ( {1,2,3,4,5,7,8}, {1,2,6,7,8}, {3,4,5,6,7,8}), A = ( {1,2,3,4,5,6,7}, {1,3,4,5,6,7}, {1,2,6,7,8}) Are M[A] and M[A ] equal? Draw their geometric representations to find out 2 Not considering how anything is labelled, there are 23 distinct geometric representations of the uniform matroid U 3,6 on a 3-vertex simplex (a triangle) (a) Find them (b) For several of them, label the elements of U 3,6 and find the corresponding presentations

28 Part II Characterizations of Transversal Matroids

29 Fundamental Transversal Matroids If a transversal matroid M has a representation on a simplex with elements of M at each vertex, then any set B that consists of exactly one element from each vertex (i) is a basis and (ii) has F = cl(b F) for all F Z(M) A fundamental transversal matroid is a matroid M having a basis B (a fundamental basis) such that F = cl(b F) for all F Z(M) Not fundamental Fundamental Fundamental transversal matroids are transversal

30 Transversal Matroids Embed in Fundamental Transversal Matroids Geometrically, a (transversal) matroid is fundamental iff it has a simplex representation that has at least one element at each vertex Each transversal matroid embeds in a fundamental transversal matroid of the same rank Group Discussion: What identifying property do the corresponding presentations have?

31 A Consequence of Inclusion/Exclusion For a fundamental transversal matroid M, a fundamental basis B, and X 1,X 2,,X n Z(M), besides r(x i ) = B X i we get r(x 1 X 2 X n ) = B (X1 X 2 X n ) r(x 1 X 2 X n ) = B (X1 X 2 X n )

32 A Consequence of Inclusion/Exclusion For a fundamental transversal matroid M, a fundamental basis B, and X 1,X 2,,X n Z(M), besides r(x i ) = B X i we get r(x 1 X 2 X n ) = B (X1 X 2 X n ) r(x 1 X 2 X n ) = B (X1 X 2 X n ) Inclusion/exclusion (PIE; dual of the more common form): C 1 C 2 C n = C j J {1,,n}( 1) J +1 j J Eg, C 1 C 2 = C 1 + C 2 C 1 C 2

33 A Consequence of Inclusion/Exclusion r(x 1 X 2 X n ) = B (X 1 X 2 X n ) r(x 1 X 2 X n ) = B (X1 X 2 X n ) PIE: C 1 C 2 C n = J {1,,n}( 1) J +1 j J C j Set C i = B X i and apply PIE: r(x 1 X 2 X n ) = ( 1) J +1 r ( ) X j J {1,,n} j J

34 An Inequality for Arbitrary Transversal Matroids If M = N A, X,Y Z(M), X = cl N (X), and Y = cl N (Y ), then X,Y Z(N) and r M (X Y ) r N (X Y ) and r M (X Y ) = r N (X Y ) Thus, a necessary condition for M to be transversal is: for all nonempty collections {X 1,X 2,,X n } Z(M), r(x 1 X 2 X n ) ( 1) J +1 r ( ) X j J {1,,n} j J

35 This Necessary Condition is Sufficient Shorthand: for F Z(M), F = X F X and F = X F X Theorem A matroid M is transversal iff for all nonempty F Z(M), r( F) ( 1) F +1 r( F ) F F (Mason, 1971; refined by Ingleton, 1977)

36 Example Time Mini Problem Session Theorem A matroid M is transversal iff for all nonempty F Z(M), r( F) ( 1) F +1 r( F ) F F b d c e a f g 5 6 Problem: Find sets F in these non-transversal rank-3 matroids for which the inequality fails

37 Outline of the Proof of the Sufficiency (1 of 3) Observation In M = M[A 1,,A r ], each complement A c i = E(M) A i is a flat Theorem Any transversal matroid has a presentation A = (A 1,,A r ) in which each complement A c i is a cyclic flat The proof applies a path-augmenting argument to show that A i x can replace A i whenever x is a coloop of M\A i Iteration gives the result

38 Outline of the Proof of the Sufficiency (2 of 3) Any transversal matroid has a presentation A = (A 1,,A r ) made up of complements of cyclic flats Recall: for any cyclic flat F of M[A], there are exactly r(f) integers i with F A i, ie, F A c i, so there are r(m) r(f) integers i with F A c i Working from cyclic flats F of higher rank towards those of lower rank, this condition yields a unique recursively-defined function β where β(f), for F Z(M), must be the multiplicity of F c in A β(f) = r(m) r(f) Y Z(M):F Y β(y )

39 Outline of the Proof of the Sufficiency (3 of 3) The multiplicity of F c in A is given recursively by β(f) = r(m) r(f) β(y ) Y Z(M):F Y Guided by this, starting with a matroid M that satisfies the inequality in the Mason/Ingleton result, recursively define β as above, show β(f) 0, let A be the set system in which, for each F Z(M), F c occurs β(f) times in A, and show that the resulting transversal matroid M[A] has the same independent sets as M, so M = M[A]

40 The Start of the Details: Sets in Presentations Observation For M = M[A 1,,A r ], each complement A c i = E(M) A i is a flat of M since r(a c i x) = r(a c i ) + 1 for all for x A i Theorem If x is a coloop of M\A i, then (A 1,,A i 1,A i x,a i+1,,a r ) is also a presentation of M

41 If x is a coloop of M\A 1, A 1 x A 2 A 3 A r x b 2 b 3 b r A = (A 1, A 2,, A r ) A = (A 1 x, A 2,,A r ) A transversal T of A with a matching that does not apply to A T = {b 2,,b r } A 1 is a partial transversal of (A 2 A 1,,A r A 1 ), so T x is also a partial transversal of (A 2 A 1,, A r A 1 ) Color the edges in a matching of T x into (A 2 A 1,, A r A 1 ) blue Look at the longest blue/red path starting at x; reindex to simplify subscripts The last vertex (shown as b 3 ) must be in A 1 T A 2 A 1 A 3 A 1 x b 2 b 3 Use the blue edges leaving b 3 for A 1 A 1 A 2 A 3 A 4 A r b 3 x b 2 b 4 b r

42 Sets in Presentations If x is a coloop of M\A i, then (A 1,,A i 1,A i x,a i+1,,a r ) is also a presentation of M Corollary For some presentation (A 1,,A r ) of M, each complement E(M) A i is a cyclic flat

43 Multiplicities of Cyclic Flats in Presentations The sum of the multiplicities must be r = r(m) If F is a cyclic flat of M = M[A 1,,A r ], then there are exactly r(f) integers i with F A i, ie, F A c i Thus, if β(y ) is multiplicity of Y c, where Y Z(M), then for each F Z(M), we have β(y ) = r(f) Y Z(M) :F Y so Y Z(M) :F Y β(y ) = r r(f)

44 The Definition of β Define β recursively: for X E(M), β(x) = r r(x) β(f) F Z(M) :X F b d c e β(a, b, c) = 1 0 = 1 β(a) = = 1 β( ) = = 0 a f g If X Z(M), then β(f) = r r(x) F Z(M) :X F

45 Example Time Mini Problem Session β(x) = r r(x) β(f) F Z(M) :X F Problem: In this example, find a set X with β(x) < 0

46 Examples of the Presentation β Gives Mini Problem Session For the uniform matroid U r,n of rank r < n on S = {1,2,,n}, we have Z(U r,n ) = {,S} Thus, β(x) = r r(x) for all X S One presentation of U r,n has S (no other sets) repeated r times β(x) = r r(x) β(f) F Z(M):X F a b i c j d e Problem: Find the presentation that β gives in this example h g f

47 The Inequality in the Mason-Ingleton Result Gives β 0 Theorem If, for every nonempty subset F of Z(M), r( F) ( 1) F +1 r( F ), F F then β(x) 0 for all X E(M)

48 The Proof Proof: For X E(M), set F(X) = {Y Z(M) : X Y } Proving β(x) 0 is equivalent to proving Y F(X) We may assume F(X) β(y ) r r(x) Some algebra we use: if K is a nonempty set, then ( 1) J +1 = 1 J : J K Upon setting K = n, this is all but the first term in ( ) ( ) ( ) ( ) ( ) n n n n n ( 1) n+1 = n

49 The Proof (continued) Now β(y ) = β(y ) Y F(X) Y F(X) F {Z F(X):Z Y } = ( 1) F +1 F F(X) Y F(X): F Y ( 1) F +1 β(y ) = ( 1) F +1 ( r r( F ) ) F F(X) = r ( 1) F +1 r( F ) F F(X) r r( F(X)) r r(x)

50 The Proof (continued) Now β(y ) = β(y ) Y F(X) Y F(X) F {Z F(X):Z Y } = ( 1) F +1 F F(X) Y F(X): F Y ( 1) F +1 β(y ) = ( 1) F +1 ( r r( F ) ) F F(X) = r ( 1) F +1 r( F ) F F(X) r r( F(X)) r r(x)

51 The Proof (continued) Now β(y ) = β(y ) Y F(X) Y F(X) F {Z F(X):Z Y } = ( 1) F +1 F F(X) Y F(X): F Y ( 1) F +1 β(y ) = ( 1) F +1 ( r r( F ) ) F F(X) = r ( 1) F +1 r( F ) F F(X) r r( F(X)) r r(x)

52 The Proof (continued) Now β(y ) = β(y ) Y F(X) Y F(X) F {Z F(X):Z Y } = ( 1) F +1 F F(X) Y F(X): F Y ( 1) F +1 β(y ) = ( 1) F +1 ( r r( F ) ) F F(X) = r ( 1) F +1 r( F ) F F(X) r r( F(X)) r r(x)

53 The Proof (conclusion) Now β(y ) = β(y ) Y F(X) Y F(X) F {Z F(X):Z Y } = ( 1) F +1 F F(X) Y F(X): F Y ( 1) F +1 β(y ) = ( 1) F +1 ( r r( F ) ) F F(X) = r ( 1) F +1 r( F ) F F(X) r r( F(X)) r r(x)

54 A Background Result Needed for the Remaining Implication Which set systems have transversals? S = {1,2,3,4} with A = ({1}, {2}, {1,2}, {3,4}) has no transversal Hall s Theorem A (finite) set system (A j : j J) has a transversal if and only if A i K for all K J i K

55 The Remaining Implication Theorem For a matroid M, if β(x) 0 for all X E(M), then M is transversal; one presentation A consists of the sets F c, for F Z(M), with F c having multiplicity β(f) Proof: Goal: show M = M[A] The sum of the multiplicities in A is r r(cl( )) = r If C is a circuit of M, then cl(c) Z(M); by how β and A are defined, cl(c) intersects only r(c) sets of A, counting multiplicity; thus, dependent sets in M are dependent in M[A]

56 Proof (conclusion) Write A as (F c 1,,F c r ) It remains to show that each basis B of M is independent in M[A], ie, A B = (F c 1 B,,F c r B) has a transversal (namely, B) By Hall s theorem, A B has a transversal iff for each j r, each union, say X, of any j sets in A B contains at least j elements Now X = i J (F c i B) for some J [r] with J = j Thus, B X i J F i, so j β(y ) r r(b X) X Y Z(M) :B X Y

57 Characterizations of Transversal Matroids Theorem For a matroid M, the following statements are equivalent: M is transversal, for every nonempty subset F of Z(M), r( F) ( 1) F +1 r( F ), (*) F F β(x) 0 for all X E(M) It suffices for (*) to hold for all collections F of incomparable sets in Z(M)

58 Maximal Presentations (A 1,,A r ) is a maximal presentation of M if M has no presentation of the form (A 1,,A i x,,a r ) with x A i If (A 1,,A r ) is a maximal presentation of M, then each set A c i is a cyclic flat of M If F is a cyclic flat of M = M[A 1,,A r ], then there are exactly r(f) integers i with F A i, ie, F A c i The recursive definition of β shows that there is only one option for the multiplicities Theorem Each transversal matroid has a unique maximal presentation (Mason, 1970) ***

59 Summary of Part 2 fundamental transversal matroids for FTMs, counting (inclusion/exclusion) gives r( F) = F F ( 1) F +1 r( F ) r( F) F F ( 1) F +1 r( F ) follows for transversal main result: this inequality characterizes transversal matroids each transversal matroid has a presentation by complements of cyclic flats {i : A i F } = r(f) for each cyclic flat F β is defined to satisfy these equalities β being nonnegative allows us to define a presentation of M by complements of cyclic flats (β(f) is the multiplicity of F c ) corollary: the maximal presentation is unique

60 Part III Lattice Path Matroids

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

62 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

63 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

64 Paths as Transversals of Set Systems Each path in P is determined by its north steps The lattice path with north steps 1, 3, 7,

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

66 Paths as Transversals of Set Systems Each path in P is determined by its north steps The lattice path with north steps 1, 3, 7, Label each north step by its position in each path it is in N 1 = {1,2,3,4}

67 Paths as Transversals of Set Systems Each path in P is determined by its north steps The lattice path with north steps 1, 3, 7, Label each north step by its position in each path it is in N 1 = {1,2,3,4} N 2 = {2,3,4,5,6}

68 Paths as Transversals of Set Systems Each path in P is determined by its north steps The lattice path with north steps 1, 3, 7, Label each north step by its position in each path it is in N 1 = {1,2,3,4} N 2 = {2,3,4,5,6} N 3 = {4,5,6,7}

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

70 Lattice Path Matroids Let P, Q, P, and N 1,N 2,,N r be as above 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 M[P,Q] for some such P and Q

71 Bases and Paths 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]

72 Bases and Paths 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]

73 Bases and Paths 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] Corollary The number of bases of M[P,Q] is P

74 Direct Sums The bases of M 1 M 2 are the sets B 1 B 2 where B i is a basis of M i M 1 M 2 M 1 M 2 Theorem The class of lattice path matroids is closed under direct sums Theorem The lattice path matroid M[P,Q] is connected if and only if P and Q intersect only at the ends

75 Spanning Circuits Theorem Connected lattice path matroids have spanning circuits

76 Spanning Circuits Theorem Connected lattice path matroids have spanning circuits

77 Spanning Circuits Theorem Connected lattice path matroids have spanning circuits

78 Spanning Circuits Theorem Connected lattice path matroids have spanning circuits Theorem An element of M[P,Q] is in a spanning circuit of M[P,Q] if and only if it is in at least two sets N i, or it is in either N 1 or N r

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

80 Deletion and Minors The bases of the deletion M\x by a non-coloop x are the bases B of M with x B x x M x x forbidden step allowed step M\x Theorem The class of lattice path matroids is closed under minors ***

81 Moral of Part 3 Sometimes subclasses of transversal matroids can have more structure than the full class (Eg, we picked up closure under minors and duals by focusing on lattice path matroids)

82 Part IV Tutte Polynomials of Lattice Path Matroids

83 The Tutte Polynomial The Tutte polynomial of M is t(m;x,y) = (x 1) r(m) r(a) (y 1) A r(a) A E(M) From the Tutte polynomial, we get chromatic and flow polynomials of graphs, weight enumerators of linear codes, information about arrangements of hyperplanes, Jones polynomials of alternating knots, the partition function of the Ising model, and information about lattice paths For (transversal) matroids, computing t(m;x,y) is #P hard Tutte polynomials of lattice path matroids are atypically accessible

84 Examples of Tutte Polynomials M 1 M 2 (x 1) 3 (the empty set) +6(x 1) 2 (the six singleton sets) +15(x 1) (the fifteen pairs) +18 (eighteen of the 3-subsets have rank 3) +2(x 1)(y 1) (the other two 3-subsets have rank 2) + 15(y 1) (the 4-element subsets) +6(y 1) 2 (the 5-element subsets) +(y 1) 3 (the entire set) which is x 3 + 3x 2 + 4x + 2xy + 4y + 3y 2 + y 3

85 Reformulation of the Tutte Polynomial via Basis Activities Linearly order E(M) The element b of a basis B is internally active for B if b = min{x : (B b) x is a basis};

86 Reformulation of the Tutte Polynomial via Basis Activities Linearly order E(M) The element b of a basis B is internally active for B if b = min{x : (B b) x is a basis}; i(b) is the number of such b

87 Reformulation of the Tutte Polynomial via Basis Activities Linearly order E(M) The element b of a basis B is internally active for B if b = min{x : (B b) x is a basis}; i(b) is the number of such b The element a B is externally active for B if a = min{x : (B a) x is a basis}; e(b) is the number of such a

88 Reformulation of the Tutte Polynomial via Basis Activities Linearly order E(M) The element b of a basis B is internally active for B if b = min{x : (B b) x is a basis}; i(b) is the number of such b The element a B is externally active for B if a = min{x : (B a) x is a basis}; e(b) is the number of such a Theorem t(m;x,y) = bases B x i(b) y e(b) (Crapo, 1967)

89 Corollaries of the Reformulation of the Tutte Polynomial Internally active: b = min{x : (B b) x is a basis} Externally active: a = min{x : (B a) x is a basis} Lemma The element b is internally active for B in M if and only if b is externally active for S B in M Corollary t(m ;x,y) = t(m;y,x)

90 The Lattice Path Interpretation of Activities Order: 1 < 2 < < m + r Internally active: b = min{x : (B b) x is a basis} Theorem The internally active elements in B correspond to the north steps of the associated path P B that are on the upper bounding path Q Q P

91 The Lattice Path Interpretation of Activities Order: 1 < 2 < < m + r Internally active: b = min{x : (B b) x is a basis} Theorem The internally active elements in B correspond to the north steps of the associated path P B that are on the upper bounding path Q b

92 The Lattice Path Interpretation of Activities Order: 1 < 2 < < m + r Internally active: b = min{x : (B b) x is a basis} Theorem The internally active elements in B correspond to the north steps of the associated path P B that are on the upper bounding path Q b b

93 The Lattice Path Interpretation of Activities Order: 1 < 2 < < m + r Internally active: b = min{x : (B b) x is a basis} Theorem The internally active elements in B correspond to the north steps of the associated path P B that are on the upper bounding path Q b

94 The Lattice Path Interpretation of Activities Order: 1 < 2 < < m + r Internally active: b = min{x : (B b) x is a basis} Theorem The internally active elements in B correspond to the north steps of the associated path P B that are on the upper bounding path Q Q Corollary e(b) is the number of east steps P B shares with the lower bounding path P P

95 Coefficients of Tutte Polynomials of Lattice Path Matroids Theorem The coefficient of x i y j in the Tutte polynomial of M[P,Q] is the number of paths in P sharing i north steps with Q and j east steps with P A basis, realized as a path, that contributes x 2 y to the Tutte polynomial x Q x y P

96 A Polynomial-Time Algorithm for Computing Tutte Polynomials of Lattice Path Matroids The recurrence to compute Tutte polynomials for lattice path matroids: x 2 + xy x 2 + xy + x + y + y 2 f(x, y) y f(x, y) f(x, y) f(x, y) + g(x, y) xf(x, y) x x + y x + y + y 2 g(x, y) f(x, y) 1 y y 2 Corollary The Tutte polynomial of a lattice path matroid can be computed in polynomial time

97 Global Summary there are many ways to view transversal matroids (set systems, bipartite graphs, matrices, simplex representations) each perspective makes certain properties apparent there are characterizations of transversal matroids that arise from appealing geometric ideas transversal matroids form an interesting, concrete class of matroids with a good bit of structure, but lacking some desirable properties (eg, closure under minors and duals) some special representations of some transversal matroids (eg, lattice path matroids) lead to more structure and more attractive properties