Another follow up...
1) Concerning commutativity, we have p_n pleth f = f pleth p_n, but, I guess,
not generally f pleth g = g pleth f. See Stanley, EC2, page 447 (Chapter 7,
Appendix 2, Def A2.6)
2) Marni Mishna's thesis
www.math.sfu.ca/~mmishna/Publications/thesis.pdf
(page 116 and 117) confirms my account. See below
Martin
--quoting-----------------------------------------------------------------
Associated to each species is a cycle index series, (in the sense of
Polya). This series makes automatic many notions linked to Polya theory. In
particular the enumeration of structures up to isomorphism. Define pn as the
power sum symmetric function. To each species F we associate a symmetric series
ZF (p1 , p2 , . . .), which is defined as follows. Definition 7.1 cycle index
series.
For any species F define its cycle index series ZF as the series in
C[[p1,p2,...]]:
m1 m2 mk
p1 p2 .AN7 .AN7 .AN7 pk
ZF (p1, p2,...) := sum sum Fix F[$B&K(B] --------------------- (7.2)
n $B&K(Bn z$B&K(B
where the value of Fix F[$B&K(B] is the number of structures of F which remain
fixed under some labelling permutation of type $B&K(B, and mk gives the
number of
parts of $B&K(B equal to k.
[...]
Another useful operation is substitution. The substitution of two species
F1 o F2, denoted (F1 o F2)[U] is formally defined as
(F1 o F2) [U ] := sum F1 [$B&P(B] .ANW prod F2
[$B&B(B],
$B&P":(BPart[U] $B&B":&P(B
where Part[U] stands for the set of partitions of U . Elements of $B&P(B are
the
blocks of the partition.
[...]
The effect on the cycle index series is best described using symmetric function
plethysm pn[g] as defined on page 29
ZF1oF2 (p1, p2, p3, ...) = ZF1 [ZF2 ] = ZF1(p1[ZF2], p2[ZF2],...). (7.5)
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