Ralf Hemmecke <[EMAIL PROTECTED]> writes:
> On 02/14/2007 09:17 AM, Martin Rubey wrote:
> >> As you know in phlethystic composition we need to "stretch" a given series.
>
> >> To say things in a more intuitive language: if f(x_1, x_2, x_3, ...) is a
> >> formal power series in infinitely many variables, the k-th stretch is
> >> f(x_{k}, x_{2k}, x_{3k}, ...).
>
> > Unfortunately, I only have a quarter of an answer, taking us into the realm
> > of
> > symmetric functions.
>
> > As you might know, the variables x_i in the cycle indicator series *should*
> > be
> > interpreted really as the i-th power sums, thus I'd prefer to write p_i. To
> > avoid confusion, I write z1, z2, z3,... for the *arguments* of these
> > symmetric
> > functions.
>
> > p_i(z1,z2,z3,...) = z1^i+z2^i+z3^i+...
> >
> > Thus, the cycle index series is a symmetric function in the z1, z2, z3, ...,
> > usually defined in terms of the power sum symmetric functions p_1, p_2,
> > p_3... Unfortunately (in my current opinion), we denote the p_i currently
> > with
> > x_i, as BLL did.
>
> Sorry, but I cannot believe what you say.
I don't understand what you cannot believe?
I just checked Stanley Enumerative Combinatorics II, Section 7.24, Enumeration
under Group Action. I give a brief account here, but I'm not firm on this
topic, currently...
Stanley defines the cycle indicator for any subset K of the symmetric group
\frac S_S as
Z_K = 1/#K \sum_{w\in K} p_{\rho(w)}
where \rho(w) is the cycle type of w and p_{i_1, i_2,...} is the product of the
power sums p_{i_j}.
He then writes:
In the traditional exposition of Poly theory [...], the power sum symmetric
functions p_i is replaced by an indeterminate t_i, and later one
substitutes p_i or a specialization of p_i for t_i.
This change of viewpoint has the following advantage: Consider the cycle
indicator for graphs with 4 vertices, G the group permuting the vertices:
Z_G = 1/24 (p1^6 + 9 p1^2 p2^2 + 8 p3^2 + 6 p2 p4)
Then, Polya theory is supposed to tell us that the "store enumerator" is the
same thing, expressing Z_G in the basis of monomial symmetric functions:
F_G = m_6 + m_51 + 2 m_42 + 3 m_33 + 2 m_411 + 4 m_321 + 6 m_222
+ 5 m 3111 + 9 m_2211 + 15 m_21111 + 30 m_111111
To obtain the generating function of nonisomorphic loopless graphs on 4
vertices, q encoding the number of edges, we can simply compute
Z_G(1,q)=F_G(1,q)=1+q+2q^2+3q^3+2q^4+q^5+q^6.
The arguments (1, q) are of course the arguments to the power sums and monomial
symmetric functions.
However, in our current notation, the arguments of Z_G are not the arguments of
the power sums, but rather the power sums themselves. So we have to write
Z_G(1+q, 1+q^2, 1+q^3, ...)
I think the main argument favouring Stanley's notation is the beautiful way to
spell out Polya's theorem: Z_G = F_G.
(Remark: BLL writes P_G for Stanley's Z_G and roughly |\Phi..| for F_G, but I
didn't work out the details)
Martin
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