1 Partition Function Integrand 
Consider the Partition Function Z of a Quantum Statistical Mechan­ 
ical System 
Z = Tr exp(-#H) . (1) 
whose hamiltonian is H (# # 1/kT , k=Boltzmann's constant 
T=absolute temperature). We may evaluate the trace over any com­ 
plete set of states; we choose the (over­)complete set of coherent states 
|z# = e -|z| 2 
|/2 # n 
(z n / # n!)a + n 
|0# (2) 
where a + is the boson creation operator satisfying [a.a + ] = 1 and for 
which the completeness or resolution of unity property is 
1 
# 
# d 2 z|z##z| = I # # dµ(z)|z##z|. (3) 
The simplest, and generic, example is the free single­boson hamiltonian 
H = #a + a for which the appropriate trace calculation is 
Z = 
1 
# 
# d 2 z#z| exp # -##a + a # |z# = 
= 1 
# 
# d 2 z#z| : exp # a + a(e -## - 1) # : |z#. (4) 
Here we have used the following well­known relation [?, ?] for the 
forgetful normal ordering operator : f(a, a + ) : which means ``normally 
order the creation and annihilation operators in f forgetting the com­ 
mutation relation [a, a + ] = 1.'' 1 
We may write the Partition Function in general as 
Z(x) = # F (x, z) dµ(z) (5) 
thereby defining the Partition Function Integrand (PFI) F (x, z). We 
have explicitly written the dependence on x # -#, the inverse tem­ 
perature, and #, the energy scale in the hamiltonian. 
1 Of course, this procedure may alter the value of the operator to which it is 
applied. 

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2 Combinatorial aspects: Bell numbers 
The generic free­boson example Eq.(5)above may be rewritten to show 
the connection with certain well­known combinatorial numbers. Writ­ 
ing y = |z| 2 and x = -##, Eq.(5) becomes 
Z = 
# 
# 0 
dy exp # y(e x 
- 1) # . (6) 
This is an integral over the classical exponential generating function 
for the Bell polynomials 
exp # y # e x 
- 1 ## = 
# 
# n=0 
B n (y) x n 
n! 
(7) 
where the Bell number is B n (1) = B(n), the number of ways of putting 
n di#erent objects into n identical containers (some may be left empty). 
Related to the Bell numbers are the Stirling numbers of the second kind 
S(n, k), which are defined as the number of ways of putting n di#er­ 
ent objects into k identical containers, leaving none empty. From the 
definition we have B(n) = # n 
k=1 S(n, k). The foregoing gives a com­ 
binatorial interpretation of the partition function integrand F (x, y) as 
the exponential generating function of the Bell polynomials. 
2.1 Graphs 
We now give a graphical representation of the Bell numbers. Consider 
labelled lines which emanate from a white dot, the origin, and finish on 
a black dot, the vertex. We shall allow only one line from each white 
dot but impose no limit on the number of lines ending on a black 
dot. Clearly this simulates the definition of S(n, k) and B(n), with 
the white dots playing the role of the distinguishable objects, whence 
the lines are labelled, and the black dots that of the indistinguishable 
containers. The identification of the graphs for 1,2 and 3 lines is given 
in Figure 1. We have concentrated on the Bell number sequence and 
its associated graphs since, as we shall show, there is a sense in which 
this sequence of graphs is generic. That is, we can represent any 
combinatorial sequence by the same sequence of graphs as in the Figure 

4 
Figure 1. Graphs for B(n), n = 1, 2, 3. 
1, with suitable vertex multipliers (denoted by the V terms in the same 
figure). Consider a general partition function 
Z = Tr exp(-#H) (8) 
where the Hamiltonian is given by H = #w(a, a + ), with w a string (= 
sum of products of positive powers) of boson creation and annihilation 
operators. The partition function integrand F for which we seek to 
give a graphical expansion, is 
Z(x) = # F (x, z) dµ(z) , (9) 

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where 
F (x, z) = #z| exp(xw)|z# = (x = -##) 
= 
# 
# 
n=0 
#z|w n 
|z# 
x n 
n! 
= 
= 
# 
# 
n=0 
W n (z) x n 
n! 
= 
= exp # # 
# n=1 
V n (z) x n 
n! 
# , (10) 
with obvious definitions of W n and V n . The sequences {W n } and {V n } 
may each be recursively obtained from the other [?]. This relates 
the sequence of multipliers {V n } of Figure 1 to the Hamiltonian of 
Eq.(8). The lower limit 1 in the V n summation is a consequence of the 
normalization of the coherent state |z#. 
3 Hopf Algebra structure 
We briefly describe the Hopf algebra L Bell which the diagrams of Figure 
1 define. 
1. Each distinct diagram is an individual basis element of L Bell ; thus 
the dimension is infinite. (Visualise each diagram in a ``box''.) 
The sum of two diagrams is simply the two boxes containing the 
diagrams. Scalar multiples are formal; for example, they may be 
provided by the V coe#cients. 
2. The identity element e is the empty diagram (an empty box). 
3. Multiplication is the juxtaposition of two diagrams within the 
same ``box''. L Bell is generated by the connected diagrams; this 
is a consequence of the Connected Graph Theorem[5]. Since we 
have not here specified an order for the juxtaposition, multipli­ 
cation is commutative. 
4. The coproduct #L Bell # L Bell × L Bell is defined by 
#(e) = e × e (unit e) 
#(x) = x × e + e × x (generator x) 
#(AB) = #(A)#(B) otherwise 

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so that # is an algebra homomorphism. 
References 
[1] A readable account may be found in Dirk Kreimer's ''Knots and Feyn­ 
man Diagrams'', Cambridge Lecture Notes in Physics, CUP (2000). 
[2] We use the graphical description of Bender, Brody and Meister, 
J.Math.Phys. 40, 3239 (1999) and arXiv:quant­ph/0604164 
[3] This approach is extended in Blasiak, Penson, Solomon, Horzela 
and Duchamp, J.Math.Phys. 46, 052110 (2005) and arXiv:quant­ 
ph/0405103 
[4] A preliminary account of the more mathematical aspects of this work 
may be found in arXiv: cs.SC/0510041 
[5] GW Ford and GE Uhlenbeck,Proc. Nat. Acad. 42, 122,1956. 

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