1. CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo

2. CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo I. The Lagrange Implicit Function Theorem and Exponential Generating Functions

3. CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo I. The Lagrange Implicit Function Theorem and Exponential Generating Functions II. A Smorgasbord of Combinatorial Identities

4. 1. Multivariate Lagrange Implicit Function Theorem

5. II. A Smorgasbord of Combinatorial Identities 1. Multivariate Lagrange Implicit Function Theorem 2. The MacMahon Master Theorem

6. II. A Smorgasbord of Combinatorial Identities 1. Multivariate Lagrange Implicit Function Theorem 2. The MacMahon Master Theorem 3. Cartier-Foata (Viennot) Heap Inversion

7. II. A Smorgasbord of Combinatorial Identities 1. Multivariate Lagrange Implicit Function Theorem 2. The MacMahon Master Theorem 3. Cartier-Foata (Viennot) Heap Inversion 4. Karlin-MacGregor/Lindstrom/Gessel-Viennot

8. II. A Smorgasbord of Combinatorial Identities 1. Multivariate Lagrange Implicit Function Theorem 2. The MacMahon Master Theorem 3. Cartier-Foata (Viennot) Heap Inversion 4. Karlin-MacGregor/Lindstrom/Gessel-Viennot 5. Kirchhoff’s Matrix Tree Theorem

9. II. A Smorgasbord of Combinatorial Identities 1. Multivariate Lagrange Implicit Function Theorem 2. The MacMahon Master Theorem 3. Cartier-Foata (Viennot) Heap Inversion 4. Karlin-MacGregor/Lindstrom/Gessel-Viennot 5. Kirchhoff’s Matrix Tree Theorem 6. The “Four-Fermion Forest Theorem” (C-J-S-S-S)

11. 1. Multivariate LIFT Commutative ring K Indeterminates and Power series in K[[u]].

12. 1. Multivariate LIFT Commutative ring K Indeterminates and Power series in K[[u]]. (a) There are unique power series in K[[x]] such that for each 1 <= j <= n.

13. 1. Multivariate LIFT (b) For these power series and for any monomial (I.J. Good, 1960)

15. 2. The MacMahon Master Theorem Special case of Multivariate LIFT in which each is a homogeneous linear form.

16. 2. The MacMahon Master Theorem Special case of Multivariate LIFT in which each is a homogeneous linear form. (MacMahon, 1915)

17. 2. The MacMahon Master Theorem This can be rephrased as….

18. 2. The MacMahon Master Theorem This can be rephrased as…. The matrix represents an endomorphism on an n-dimensional vector space V.

19. 2. The MacMahon Master Theorem This can be rephrased as…. The matrix represents an endomorphism on an n-dimensional vector space V. There are induced endomorphisms on the symmetric powers of V, and on the exterior powers of V.

20. 2. The MacMahon Master Theorem The traces of these induced endomorphisms satisfy

21. 2. The MacMahon Master Theorem The traces of these induced endomorphisms satisfy

22. 2. The MacMahon Master Theorem By the MacMahon Master Theorem… This is called the “Boson-Fermion Correspondence”

23. 2. The MacMahon Master Theorem By the MacMahon Master Theorem… This is called the “Boson-Fermion Correspondence” (Garoufalidis-Le-Zeilberger, 2006) “quantum” MacMahon Master Theorem.

25. 3. Cartier-Foata/Viennot Heap Inversion Another example of the Boson-Fermion Correspondence arising from symmetric functions…. Countably many indeterminates

26. 3. Cartier-Foata/Viennot Heap Inversion Another example of the Boson-Fermion Correspondence arising from symmetric functions…. Countably many indeterminates Elementary symmetric functions

27. 3. Cartier-Foata/Viennot Heap Inversion Another example of the Boson-Fermion Correspondence arising from symmetric functions…. Countably many indeterminates Elementary symmetric functions Complete symmetric functions

28. 3. Cartier-Foata/Viennot Heap Inversion Generating functions…

29. 3. Cartier-Foata/Viennot Heap Inversion Generating functions…

30. 3. Cartier-Foata/Viennot Heap Inversion Generating functions… Clearly

31. 3. Cartier-Foata/Viennot Heap Inversion Let G=(V,E) be a simple graph. A subset S of V is stable provided that no edge of G has both ends in S.

32. 3. Cartier-Foata/Viennot Heap Inversion Let G=(V,E) be a simple graph. A subset S of V is stable provided that no edge of G has both ends in S. Introduce indeterminates The stable set enumerator of G is

33. 3. Cartier-Foata/Viennot Heap Inversion Let G=(V,E) be a simple graph. A subset S of V is stable provided that no edge of G has both ends in S. Introduce indeterminates The stable set enumerator of G is (Partition function of a zero-temperature lattice gas on G with repulsive nearest-neighbour interactions.)

34. 3. Cartier-Foata/Viennot Heap Inversion Let G=(V,E) be a simple graph. Introduce indeterminates Say that these commute only for non-adjacent vertices: if and only if

35. 3. Cartier-Foata/Viennot Heap Inversion Let G=(V,E) be a simple graph. Introduce indeterminates Say that these commute only for non-adjacent vertices: if and only if Let be the set of all finite strings of vertices, modulo the equivalence relation generated by these commutation relations.

36. 3. Cartier-Foata/Viennot Heap Inversion (Cartier-Foata, 1969) This identity is valid for power series with merely partially commutative indeterminates, as above.

37. 3. Cartier-Foata/Viennot Heap Inversion (Cartier-Foata, 1969) This identity is valid for power series with merely partially commutative indeterminates, as above. (There are several variations and generalizations of this.) (Viennot, 1986) (Krattenthaler, preprint)

39. 4. Karlin-MacGregor/Lindstrom/Gessel-Viennot Let G=(V,E) be a finite connected acyclic directed graph properly drawn in the plane.

40. 4. Karlin-MacGregor/Lindstrom/Gessel-Viennot Let G=(V,E) be a finite connected acyclic directed graph properly drawn in the plane. Let each edge e be weighted by a value w(e) in some commutative ring K.

41. 4. Karlin-MacGregor/Lindstrom/Gessel-Viennot Let G=(V,E) be a finite connected acyclic directed graph properly drawn in the plane. Let each edge e be weighted by a value w(e) in some commutative ring K. For a path P, let w(P) be the product of the weights of the edges of P.

42. 4. Karlin-MacGregor/Lindstrom/Gessel-Viennot Let G=(V,E) be a finite connected acyclic directed graph properly drawn in the plane. Let each edge e be weighted by a value w(e) in some commutative ring K. For a path P, let w(P) be the product of the weights of the edges of P. Fix vertices in that cyclic order around the boundary of the infinite face of G.

43. 4. Karlin-MacGregor/Lindstrom/Gessel-Viennot Let G=(V,E) be a finite connected acyclic directed graph properly drawn in the plane. Let each edge e be weighted by a value w(e) in some commutative ring K. For a path P, let w(P) be the product of the weights of the edges of P. Fix vertices in that cyclic order around the boundary of the infinite face of G. Let be the generating function for all (directed) paths from A_i to Z_j.

44. 4. Karlin-MacGregor/Lindstrom/Gessel-Viennot

45. The generating function for the set of all k-tuples of paths such that * the paths P_i are internally vertex-disjoint * each P_i goes from A_i to Z_i is

46. 4. Karlin-MacGregor/Lindstrom/Gessel-Viennot The generating function for the set of all k-tuples of paths such that * the paths P_i are internally vertex-disjoint * each P_i goes from A_i to Z_i is

47. 4. Karlin-MacGregor/Lindstrom/Gessel-Viennot Application: vertical edges get weight 1. horizontal edges (a,b)—(a+1,b) get weight x_b

48. 4. Karlin-MacGregor/Lindstrom/Gessel-Viennot The generating function for all paths from to is a complete symmetric function

49. 4. Karlin-MacGregor/Lindstrom/Gessel-Viennot The generating function for all paths from to is a complete symmetric function The path shown is coded by the sequence 2 2 4 7 7

50. 4. Karlin-MacGregor/Lindstrom/Gessel-Viennot Sets of vertex-disjoint paths are encoded by tableaux : 1 1 3 6 2 2 4 7 7 3 5 5 8

51. 4. Karlin-MacGregor/Lindstrom/Gessel-Viennot Sets of vertex-disjoint paths are encoded by tableaux : 1 1 3 6 2 2 4 7 7 3 5 5 8 The generating function for tableaux of a given shape is a symmetric function… skew Schur function

52. 4. Karlin-MacGregor/Lindstrom/Gessel-Viennot (dual) Jacobi-Trudy Formula * When these correspond to the irreducible representations of the symmetric groups.

53. 4. Karlin-MacGregor/Lindstrom/Gessel-Viennot (dual) Jacobi-Trudy Formula * When these correspond to the irreducible representations of the symmetric groups. * They are the minors of “generic” Toeplitz matrices.

55. 5. Kirchhoff’s Matrix Tree Theorem Let G=(V,E) be a finite connected (multi-)graph.

56. 5. Kirchhoff’s Matrix Tree Theorem Let G=(V,E) be a finite connected (multi-)graph. Direct each edge e with ends v and w arbitrarily: Either v—e  w or w—e  v.

57. 5. Kirchhoff’s Matrix Tree Theorem Let G=(V,E) be a finite connected (multi-)graph. Direct each edge e with ends v and w arbitrarily: Either v—e  w or w—e  v. Define a signed incidence matrix of G to be the V-by-E matrix D with entries

58. 5. Kirchhoff’s Matrix Tree Theorem Fix indeterminates

59. 5. Kirchhoff’s Matrix Tree Theorem Fix indeterminates Let Y be the E-by-E diagonal matrix

60. 5. Kirchhoff’s Matrix Tree Theorem Fix indeterminates Let Y be the E-by-E diagonal matrix The weighted Laplacian matrix of G is

61. 5. Kirchhoff’s Matrix Tree Theorem A graph

62. 5. Kirchhoff’s Matrix Tree Theorem A signed incidence matrix for it

63. 5. Kirchhoff’s Matrix Tree Theorem Its weighted Laplacian matrix

64. 5. Kirchhoff’s Matrix Tree Theorem Fix indeterminates Let Y be the E-by-E diagonal matrix The weighted Laplacian matrix of G is Fix any “ground vertex”

65. 5. Kirchhoff’s Matrix Tree Theorem Fix indeterminates Let Y be the E-by-E diagonal matrix The weighted Laplacian matrix of G is Fix any “ground vertex” Let be the submatrix of L obtained by deleting the row and the column indexed by

66. 5. Kirchhoff’s Matrix Tree Theorem With the notation above… where the summation is over the set of all spanning trees of G.

67. 5. Kirchhoff’s Matrix Tree Theorem With the notation above… where the summation is over the set of all spanning trees of G. Proof uses the Binet-Cauchy determinant identity and…

68. 5. Kirchhoff’s Matrix Tree Theorem Key Lemma: Let and with

69. 5. Kirchhoff’s Matrix Tree Theorem Key Lemma: Let and with Let M be the square submatrix of D obtained by * deleting rows indexed by vertices in R, and * keeping only columns indexed by edges in S.

70. 5. Kirchhoff’s Matrix Tree Theorem Key Lemma: Let and with Let M be the square submatrix of L obtained by * deleting rows indexed by vertices in R, and * keeping only columns indexed by edges in S. Then if (V,S) is a forest in which each tree has exactly one vertex in R, and otherwise

71. 5. Kirchhoff’s Matrix Tree Theorem With the notation above… where the summation is over the set of all spanning forests F of G such that each component of F contains exactly one vertex in R. “Shorthand” notation:

72. 5. Kirchhoff’s Matrix Tree Theorem With the notation above… where the summation is over the set of all spanning forests F of G

73. 5. Kirchhoff’s Matrix Tree Theorem With the notation above… where the summation is over the set of all spanning forests F of G But… we really want a formula without the multiplicities on the RHS….

75. 6. The “Four-Fermion Forest Theorem” of C-J-S-S-S Caracciolo-Jacobsen-Saleur-Sokal-Sportiello (2004) The generating function for spanning forests of G is

76. 6. The “Four-Fermion Forest Theorem” of C-J-S-S-S “Shorthand” notation The greek letters stand for fermionic (anticommuting) variables. et cetera in particular

77. 6. The “Four-Fermion Forest Theorem” of C-J-S-S-S “Shorthand” notation The greek letters stand for fermionic (anticommuting) variables. is an operator – it means keep track only of terms in which each variable occurs exactly once, counting each such term with an appropriate sign.

78. 6. The “Four-Fermion Forest Theorem” of C-J-S-S-S For any square matrix M

79. 6. The “Four-Fermion Forest Theorem” of C-J-S-S-S For any square matrix M “Shorthand” notation

80. 6. The “Four-Fermion Forest Theorem” of C-J-S-S-S For any square matrix M Compare with C-J-S-S-S:

81. 6. The “Four-Fermion Forest Theorem” of C-J-S-S-S “Traditionally” each vertex gets a commuting (bosonic) indeterminate

82. 6. The “Four-Fermion Forest Theorem” of C-J-S-S-S “Traditionally” each vertex gets a commuting (bosonic) indeterminate In C-J-S-S-S this has two anticommuting (fermionic) “superpartners”

83. 6. The “Four-Fermion Forest Theorem” of C-J-S-S-S “Traditionally” each vertex gets a commuting (bosonic) indeterminate In C-J-S-S-S this has two anticommuting (fermionic) “superpartners” and the boson is “integrated out”

84. 6. The “Four-Fermion Forest Theorem” of C-J-S-S-S “Traditionally” each vertex gets a commuting (bosonic) indeterminate In C-J-S-S-S this has two anticommuting (fermionic) “superpartners” and the boson is “integrated out”

85. 6. The “Four-Fermion Forest Theorem” of C-J-S-S-S “Traditionally” each vertex gets a commuting (bosonic) indeterminate In C-J-S-S-S this has two anticommuting (fermionic) “superpartners” and the boson is “integrated out” The integral is interpreted combinatorially, some very pretty sign-cancellations occur, and only the forests survive, each exactly once.

87. I believe there is a department of mind conducted independent of consciousness, where things are fermented and decocted, so that when they are run off they come clear. -- James Clerk Maxwell