CSM Week 1: Introductory CrossDisciplinary Seminar

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
II. A Smorgasbord of Combinatorial Identities

• 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. Kirchhoffs 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. Kirchhoffs Matrix Tree Theorem
• 6. The Four-Fermion Forest Theorem
  (C-J-S-S-S)

10
(No Transcript)
11
1. Multivariate LIFT

• Commutative ring K
• Indeterminates
  and
• Power series
  in Ku.

12
1. Multivariate LIFT

• Commutative ring K
• Indeterminates
  and
• Power series
  in Ku.
• (a) There are unique power series
  in Kx
• such that
  for each 1 lt j lt n.

13
1. Multivariate LIFT

• (b) For these power series and
  for any monomial
• (I.J. Good, 1960)

14
(No Transcript)
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.

24
(No Transcript)
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)

38
(No Transcript)
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
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

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)(a1,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.
54
(No Transcript)
55
5. Kirchhoffs Matrix Tree Theorem

• Let G(V,E) be a finite connected
  (multi-)graph.

56
5. Kirchhoffs Matrix Tree Theorem

• Let G(V,E) be a finite connected
  (multi-)graph.
• Direct each edge e with ends v and w
  arbitrarily
• Either ve?w or we?v.

57
5. Kirchhoffs Matrix Tree Theorem

• Let G(V,E) be a finite connected
  (multi-)graph.
• Direct each edge e with ends v and w
  arbitrarily
• Either ve?w or we?v.
• Define a signed incidence matrix of G to be the
• V-by-E matrix D with entries

58
5. Kirchhoffs Matrix Tree Theorem

• Fix indeterminates

59
5. Kirchhoffs Matrix Tree Theorem

• Fix indeterminates
• Let Y be the E-by-E diagonal matrix

60
5. Kirchhoffs Matrix Tree Theorem

• Fix indeterminates
• Let Y be the E-by-E diagonal matrix
• The weighted Laplacian matrix of G is

61
5. Kirchhoffs Matrix Tree Theorem

• A graph

62
5. Kirchhoffs Matrix Tree Theorem

• A signed incidence matrix for it

63
5. Kirchhoffs Matrix Tree Theorem

• Its weighted Laplacian matrix

64
5. Kirchhoffs 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. Kirchhoffs 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. Kirchhoffs Matrix Tree Theorem

• With the notation above
• where the summation is over the set of all
• spanning trees of G.

67
5. Kirchhoffs 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. Kirchhoffs Matrix Tree Theorem

• Key Lemma
• Let and
  with

69
5. Kirchhoffs 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. Kirchhoffs 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. Kirchhoffs 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. Kirchhoffs Matrix Tree Theorem

• With the notation above
• where the summation is over the set of all
  spanning forests F of G

73
5. Kirchhoffs 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.

74
(No Transcript)
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.

86
(No Transcript)
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

88
(No Transcript)