Graphes Combinatoires et Th

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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
lACFAS, 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 Ln0 for ngt2. 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 n3, 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 is well known
(see Comtet for example) Thus, using what
has been said in the beginning, with
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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 ...
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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
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.
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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
C.M. Bender, D.C. Brody, B.K.Meister , Quantum
Field Theory of Partitions, J.Math. Phys. 40,
3239 (1999)
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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
satisfying the usual
In this elementary case there are 12 terms f 3
8 terms f f 4 1 term f2 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)
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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)
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From now on, we will denote ub (raising
operator) and db (lowering op.) satisfying
d,u1. With wud, 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 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)
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With wudu, one has the following matrix
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With wudduu, 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 --gt 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, 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 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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