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. And the reason would be
Appendix 1 of BLL which says something about Polya theory. To me that
looks like a cycle index polynomial/series is a bit more abstract by
*not* specializing to sums of powers.
> Now p_{i*k} is really the plethystic substitution of p_i into p_k, and of
> course, also of p_k into p_i.
Also \cite{Kerber:AlgebraicCombinatorics:1991} defines a general
"plethysm of arbitrary polynomials". And as a definition it is not
connected to power sums or such.
p(y_1,y_2, ..., y_n) pleth q(z_1, z_2, ..., z_k) :=
q(
p(y_{1*1}, y_{1*2}, ..., y_{1*n}),
p(y_{2*1}, y_{2*2}, ..., y_{2*n}),
...
p(y_{k*1}, y_{k*2}, ..., y_{k*n}),
)
(Interestingly, that is actually substution of p into q.)
> Thus, if I'm not mistaken, stretching a symmetric function is the same as
> plethistically substituting p_k.
Or in the words of the definition above, my "k-stretching" of a
polynomial q is nothing else than
q pleth z_k
But of course it makes no sense to define my stretching in terms of
plethysm.
> I think that "stretch" is quite ok. Hornegger and Pirastu used "transform",
> which doesn't say half as much...
Good. If nobody else complains, I'll leave it "stretch"ed.
Ralf
@Book{Kerber:AlgebraicCombinatorics:1991,
author = {Adalbert Kerber},
title = {Algebraic Combinatorics via Finite Group Actions},
publisher = {Bibliographisches Institut},
address = {Mannheim},
year = 1991
}
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