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Graphes Combinatoires et Théorie Quantique des Champs

Gérard Duchamp, Université de Rouen, France

Collaborateurs :

Karol Penson, Université de Paris VI, France

Allan Solomon, Open University, Angleterre

Pawel Blasiak, Instit. of Nucl. Phys., Cracovie, Pologne

Andrzej Horzela, Instit. of Nucl. Phys., Cracovie, Pologne

Congrès de l’ACFAS, 11 mai 2004

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Content of talk A formula from QFT giving the Hadamard

product of two EGFs Development in case F(0)=G(0)=1 Expression with (Feynman-Bender and al.)

diagrams Link with packed matrices Back to physics : One parameter groups of substitutions and normal ordering of boson strings

(continuous case)Substitutions and the « exponential formula »

(discrete case) Lie-Riordan group

Conclusion

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The Hadamard product of two sequences

is given by the pointwise product

We can at once transfer this law on EGFs by

but, here, as

we get

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In case we can set

if, for example, the Ln are (non-negative) integers, F(y) is the EGF of set-partitions (see the talk of M. Rosas yesterday) which k-blocks can be coloured with Lk different

colours.As an example, let us take L1, L2 and Ln=0 for n>2. Then the objects of size n are the set-partitions of a n-set in singletons and pairs having respectively L1 and L2 colours allowed

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For n=3, we have two types : the type (three possibilities without the colours, on the left) and the type (one possibility without the colours, on the right).

It turns out that, with the colours, we have

which agrees with the computation.

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In general, we adopt the denotation

for the type of a (set) partition which means that there are a1 singletons a2 pairs a3 3-blocks a4 4-blocks and so on.

The number of set partitions with type as above iswell known (see Comtet for example)

Thus, using what has been said in the beginning, with

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Now, one can count in another way the expressionnumpart()numpart(), remarking that this is the number of couples of set partitions (P1,P2) with type(P1)=, type(P2)=. But every couple of partitions (P1,P2) has an intersection matrix ...

one has

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{1,5} {2} {3,4,6}

{1,2} 1 1 0

{3,4} 0 0 2

{5,6} 1 0 1

{1,5} {1,2}

{2} {3,4}

{3,4,6} {5,6}

Feynman-Bender (& al.)diagram

Remark: Juxtaposition of diagrams amounts to do the blockdiagonal product of the corresponding matrices which are then indexed by the product of set partitions defined by M. Rosas yesterday.

Packed matrixsee NCSFVI(GD, Hivert, and Thibon)

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Now the product formula for EGFs reads

The main interest of this new form is that we can impose rules on the counted graphs !

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Some Model Graphs*2. Lines and Vertices EX: 4 lines

(V2)2

Single and double

*C.M. Bender, D.C. Brody, B.K.Meister ,Quantum Field Theory of Partitions, J.Math. Phys. 40, 3239 (1999)

V1*V3

Single

(V1)4

Singles

V4

Quadruple

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12

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Back to physics: One parameter groups of substitutions and normal ordering of boson strings (continuous case)

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Fermion Normal Ordering Problem*

7654321 fffffff

ijijji ffff satisfying the usual

*Combinatorics and Fermion Algebra, AM Navon, Il Nuovo Cimento 16B,324(1973)

In this elementary case there are 12 terms f 3

8 terms f+ f 4 1 term f+2 f 5

The numbers 1,2, 12,.. are combinatorial in origin (see Navon reference)

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Boson Normal Ordering Problem*

1bb b b satisfying

*Combinatorial Aspects of Boson Algebra, J Katriel, Lett. Nuovo Cimento 10,565(1974)

The integers S(n,k) are the Stirling Numbers of the Second Kind.

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( ) ( , )n

n k k

k

b b S n k b b

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From now on, we will denote u=b+ (raising operator) and d=b (lowering op.)satisfying [d,u]=1.

With w=ud, one has the Stirling matrix

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In this way, two parameters families of new Bell and Stirling numbers could be defined by means of the normal ordering

with rs, and

( ),[( ) ] ( ) ( , )( )

nsr s n n r s k k

r sk s

a a a S n k a a

, ,( ) ( , )ns

r s r sks

B n S n k

see for example, P Blasiak, KA Penson and AI Solomon,The Boson Normal Ordering Problem and Generalized Bell Numbers,Annals of Combinatorics 7, 127 (2003)

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With w=udu, one has the following matrix

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With w=udduu, one gets

Each of these matrices are row-finite and induce a sequence transformation just by multiplication on the left and they form an algebra.

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With and the (infinite) matrix ,the sequence is given by

and the transformation induced over the generating series is f --> g such that

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We can observe that, if there is only one derivative in the word w the matrix is a matrix of substitution with prefactor function i.e. a transformation of the type

this is due to the fact that we can represent u,d by operators over the functions on the line (Bargmann-Fock): multiplication by x and differentiation. The resulting operator being either a vector field or the conjugate of a vector field by an automorphism. Let us compute, for example the substitution corresponding to

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On gets the first special cases and some others

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Substitutions and the « exponential formula » (discrete case)

Well known to enumerative combinatorists: (For certain classes of graphs)If C(x) is the EGF of CONNECTED graphs, thenexp(C(x)) is the EGF of all graphs. This implies that the matrix M(n,k)=number of graphs with n vertices and having k connected components is the matrix of a substitution. One can prove that if M is such a matrix (with identity diagonal), then all its powers (positive negative and fractional) are substitution matrices and form a one-parameter group of substitutions, thus coming from a vector field on the line which can be computed. But no nice combinatorics seems to emerge.

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Conclusion

• We have, following Bender and al., given a « coupled » decomposition of the product formula. This can be used to give sections of EGFs (a non-trivial problem, trisection of Hermite EGF, by Ira Gessel and al. has been obtained very recently)

• Continuous and Discrete exponentials arising from physics and combinatorics have been presented. Remains some problems as, for example, a nice combinatorial description of the (existing) one-parameter groups associated to a substitution (say, to begin with, the Stirling substitution, which seems to induce what I could call a « Lambert phenomenon »).