Graphes Combinatoires et Théorie Quantique des Champs

1. 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

2. 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

3. 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

4. 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 differentcolours. 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

5. 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.

6. In general, we adopt the denotation for the type of a (set) partition which means that there are a1singletons a2pairs a33-blocks a44-blocks and so on. The number of set partitions with type  as above is well known (see Comtet for example) Thus, using what has been said in the beginning, with

7. one has Now, one can count in another way the expression numpart()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 ...

8. {1,5} {2} {3,4,6} {1,2} 1 1 0 {3,4} 0 0 2 {5,6} 1 0 1 Packed matrix see NCSFVI (GD, Hivert, and Thibon) {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.

9. Now the product formula for EGFs reads The main interest of this new form is that we can impose rules on the counted graphs !

10. V1*V3 Single (V2)2 Single and double V4 Quadruple (V1)4 Singles Some Model Graphs* 2. Lines and Vertices EX: 4 lines *C.M. Bender, D.C. Brody, B.K.Meister , Quantum Field Theory of Partitions, J.Math. Phys. 40, 3239 (1999)

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

14. Fermion Normal Ordering Problem* satisfying the usual In this elementary case there are 12 terms f 3 8 terms f+ f 4 1 termf+2 f 5 The numbers 1,2, 12,.. are combinatorial in origin (see Navon reference) *Combinatorics and Fermion Algebra, AM Navon, Il Nuovo Cimento 16B,324(1973)

15. Boson Normal Ordering Problem* satisfying The integers S(n,k) are the Stirling Numbers of the Second Kind. *Combinatorial Aspects of Boson Algebra, J Katriel, Lett. Nuovo Cimento 10,565(1974)

16. 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

17. In this way, two parameters families of new Bell and Stirling numbers could be defined by means of the normal ordering with rs, and 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)

18. With w=udu, one has the following matrix

19. 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.

20. With and the (infinite) matrix , the sequence is given by and the transformation induced over the generating series is f --> g such that

21. 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

22. On gets the first special cases and some others

23. 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, then exp(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 connectedcomponents 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.

24. 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 beenpresented. 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 »).