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CSM Week 1: Introductory Cross-Disciplinary Seminar
Combinatorial Enumeration
Dave WagnerUniversity of Waterloo
CSM Week 1: Introductory Cross-Disciplinary Seminar
Combinatorial Enumeration
Dave WagnerUniversity of Waterloo
I. The Lagrange Implicit Function Theorem and Exponential Generating Functions
CSM Week 1: Introductory Cross-Disciplinary Seminar
Combinatorial Enumeration
Dave WagnerUniversity of Waterloo
I. The Lagrange Implicit Function Theorem and Exponential Generating Functions
II. A Smorgasbord of Combinatorial Identities
II. A Smorgasbord of Combinatorial Identities
1. Multivariate Lagrange Implicit Function Theorem
II. A Smorgasbord of Combinatorial Identities
1. Multivariate Lagrange Implicit Function Theorem
2. The MacMahon Master Theorem
II. A Smorgasbord of Combinatorial Identities
1. Multivariate Lagrange Implicit Function Theorem
2. The MacMahon Master Theorem
3. Cartier-Foata (Viennot) Heap Inversion
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
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
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)
1. Multivariate LIFT
Commutative ring K
Indeterminates and
Power series in K[[u]].
),...,,( 21 nuuuu ),...,,( 21 nxxxx
)(),...,(),( 1 uuu nGGF
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.
),...,,( 21 nuuuu ),...,,( 21 nxxxx
)(),...,(),( 1 uuu nGGF
)(xjR
),...,( 1 njjj RRGxR
1. Multivariate LIFT
(b) For these power series and for any monomial
(I.J. Good, 1960)
j
i
i
jijn u
G
G
uFRRF
)(
)(det)()(][),...,(][ 1
u
uuGuux ααα
n
nxxx ...21
21αx
)(xjR
2. The MacMahon Master Theorem
Special case of Multivariate LIFT in which each
is a homogeneous linear form.
ninii ucucG ...)( 11u
2. The MacMahon Master Theorem
Special case of Multivariate LIFT in which each
is a homogeneous linear form.
(MacMahon, 1915)
ninii ucucG ...)( 11u
ijiij
n
inini cxxcxc
i
det
1][...][
111
αα xx
2. The MacMahon Master Theorem
This can be rephrased as….
ijij
n
inini cxcxc
i
det
1...][
111
0α
αx
2. The MacMahon Master Theorem
This can be rephrased as….
The matrix represents an endomorphism
on an n-dimensional vector space V.
ijij
n
inini cxcxc
i
det
1...][
111
0α
αx
)( ijcC
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.
ijij
n
inini cxcxc
i
det
1...][
111
0α
αx
)( ijcC
CS m
Cm
2. The MacMahon Master Theorem
The traces of these induced endomorphisms satisfy
m
n
inini
mi
xcxcCtrS|| 1
11 ...][α
αx
2. The MacMahon Master Theorem
The traces of these induced endomorphisms satisfy
m
n
inini
mi
xcxcCtrS|| 1
11 ...][α
αx
mmn
m
mijij TCtrTc
0
)1(det
2. The MacMahon Master Theorem
By the MacMahon Master Theorem…
This is called the “Boson-Fermion Correspondence”
1
00
)1(
mm
n
m
m
m
mm TCtrTCtrS
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.
1
00
)1(
mm
n
m
m
m
mm TCtrTCtrS
3. Cartier-Foata/Viennot Heap Inversion
Another example of the Boson-Fermion Correspondence
arising from symmetric functions….
Countably many indeterminates,...),( 21 xxx
3. Cartier-Foata/Viennot Heap Inversion
Another example of the Boson-Fermion Correspondence
arising from symmetric functions….
Countably many indeterminates
Elementary symmetric functions
k
kiii
iiik xxxe...21
21...)(x
,...),( 21 xxx
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
k
kiii
iiik xxxe...21
21...)(x
k
kiii
iiik xxxh...21
21...)(x
,...),( 21 xxx
3. Cartier-Foata/Viennot Heap Inversion
Generating functions…
10
1)()(i
ik
kk TxTeTE x
3. Cartier-Foata/Viennot Heap Inversion
Generating functions…
10
1)()(i
ik
kk TxTeTE x
10 1
1)()(
i ik
kk Tx
ThTH x
3. Cartier-Foata/Viennot Heap Inversion
Generating functions…
Clearly
10
1)()(i
ik
kk TxTeTE x
10 1
1)()(
i ik
kk Tx
ThTH x
)(
1)(
TETH
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.
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
}:{ Vvxv x
)(
);(stableS
SGZ xx
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.)
}:{ Vvxv x
)(
);(stableS
SGZ xx
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
}:{ Vvxv x
vwwv xxxx Ewv },{
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.
}:{ Vvxv x
vwwv xxxx Ewv },{
/*V
3. Cartier-Foata/Viennot Heap Inversion
(Cartier-Foata, 1969)
This identity is valid for power series with merely partially commutative indeterminates, as above.
/* );(
1
V GZ
xx
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)
/* );(
1
V GZ
xx
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
Let G=(V,E) be a finite connected acyclic directed graph properly drawn in the plane.
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.
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.
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.
1121 ,...,,,...,, ZZZAAA kkk
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.
1121 ,...,,,...,, ZZZAAA kkk
ji ZAP
ji PwZAM:
)(),(
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
1A
2A
kA
1Z
1kZ
kZ
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_iis
),...,,( 21 kPPP
),(det)()...()(),...,,(
21
21
jiPPP
k ZAMPwPwPwk
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_iis
),...,,( 21 kPPP
),(det)()...()(),...,,(
21
21
jiPPP
k ZAMPwPwPwk
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
Application:
vertical edges get weight 1.
horizontal edges (a,b)—(a+1,b) getweight x_b
1
2
3
4
5
6
7
8
x
x
x
x
x
x
x
x
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
1
2
3
4
5
6
7
8
x
x
x
x
x
x
x
x
)1,(a
),( ka
The generating functionfor all paths from to
is a complete symmetricfunction
)1,(a ),( ka
)(xkh
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
1
2
3
4
5
6
7
8
x
x
x
x
x
x
x
x
)1,(a
),( ka
The generating functionfor all paths from to
is a complete symmetricfunction
The path shown is codedby the sequence2 2 4 7 7
)1,(a ),( ka
)(xkh
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
1
2
3
4
5
6
7
8
x
x
x
x
x
x
x
x
Sets of vertex-disjointpaths are encoded bytableaux : 1 1 3 62 2 4 7 73 5 5 8
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
1
2
3
4
5
6
7
8
x
x
x
x
x
x
x
x
Sets of vertex-disjointpaths are encoded bytableaux : 1 1 3 62 2 4 7 73 5 5 8
The generating function fortableaux of a given shapeis a symmetric function…
skew Schur function )(/ xs
4. Karlin-MacGregor/Lindstrom/Gessel-Viennot
(dual) Jacobi-Trudy Formula
* When these correspond to the irreducible
representations of the symmetric groups.
)(det)(/ xxji jihs
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.
)(det)(/ xxji jihs
5. Kirchhoff’s Matrix Tree Theorem
Let G=(V,E) be a finite connected (multi-)graph.
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—ew or w—ev.
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—ew or w—ev.
Define a signed incidence matrix of G to be theV-by-E matrix D with entries
otherwise
ev
ve
Dve
0
1
1
5. Kirchhoff’s Matrix Tree Theorem
Fix indeterminates
}:{ Eeye y
5. Kirchhoff’s Matrix Tree Theorem
Fix indeterminates
Let Y be the E-by-E diagonal matrix
}:{ Eeye y
):( EeydiagY e
5. Kirchhoff’s Matrix Tree Theorem
Fix indeterminates
Let Y be the E-by-E diagonal matrix
The weighted Laplacian matrix of G is
}:{ Eeye y
):( EeydiagY e
*DYDL
5. Kirchhoff’s Matrix Tree Theorem
A graph
5. Kirchhoff’s Matrix Tree Theorem
:= G
-1 0 0 -1 -1 0 0 0 0 -1 -1
1 0 -1 0 0 -1 0 0 0 0 0
0 0 0 0 0 1 -1 0 -1 0 1
0 -1 1 1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 -1 1 0 0
0 1 0 0 0 0 1 1 0 1 0
A signed incidence matrix for it
5. Kirchhoff’s Matrix Tree Theorem
y1 y4 y5 y10 y11 y1
y11 y4
y5 y10
y1 y1 y3 y6
y6 y3 0 0
y11 y6
y6 y7 y9 y11 0 y9 y7
y4 y3 0 y2 y3 y4 0 y2
y5 0 y9 0 y5 y8 y9 y8
y10 0 y7 y2
y8 y2 y7 y8 y10
Its weighted Laplacian matrix
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”
}:{ Eeye y
):( EeydiagY e
*DYDL
Vv 0
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
}:{ Eeye y
):( EeydiagY e
*DYDL
Vv 0
)|( 00 vvL
0v
5. Kirchhoff’s Matrix Tree Theorem
With the notation above…
where the summation is over the set of all spanning trees of G.
T Te
eyvvL )|(det 00
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…
T Te
eyvvL )|(det 00
5. Kirchhoff’s Matrix Tree Theorem
Key Lemma:
Let and with ES VR )(#)(#)(# VRS
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.
ES VR )(#)(#)(# VRS
R
S
M
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
ES VR )(#)(#)(# VRS
1det M
0det M
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:
F
FRRL y)|(det
Fe
eF yy
5. Kirchhoff’s Matrix Tree Theorem
With the notation above…
where the summation is over the set of all spanning forests F of G
F
F FintreeT
TVLI y
:
)(#)det(
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….
F
F FintreeT
TVLI y
:
)(#)det(
???F
Fy
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
Eijejjiie
F
F yLId ψθψθy )(exp)(
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
nnddddddd ...)( 2211ψθ
ijji
02 i
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.
nnddddddd ...)( 2211ψθ
)(ψθd
6. The “Four-Fermion Forest Theorem” of C-J-S-S-S
For any square matrix M
)exp()()det( ψθψθ MdM
6. The “Four-Fermion Forest Theorem” of C-J-S-S-S
For any square matrix M
“Shorthand” notation
)exp()()det( ψθψθ MdM
j
n
i
n
jijimM
1 1
ψθ
6. The “Four-Fermion Forest Theorem” of C-J-S-S-S
For any square matrix M
Compare with C-J-S-S-S:
)exp()()det( ψθψθ MdM
Eijejjiie
F
F yLId ψθψθy )(exp)(
6. The “Four-Fermion Forest Theorem” of C-J-S-S-S
“Traditionally” each vertex gets a commuting (bosonic) indeterminate vx
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”
vx
v v
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”
vx
v v
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”
v v
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.
v v
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