Dear Ralf,
Ralf Hemmecke <[EMAIL PROTECTED]> writes:
> 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.
In fact, the reason for *should* is not entirely clear to me. I vaguely
remember that it comes from the usual cycle indicator, but since I find this
stuff (enumeration under group action) quite difficult, I forgot the reasons
again. (I knew last autumn...)
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.
Thus, if I'm not mistaken, stretching a symmetric function is the same as
plethistically substituting p_k. So maybe you could call it powerSubst or
something, but this doesn't make things clearer than stretch, I guess.
> Of course, I have "invented" the term "stretch". Does somebody know a better
> name for that operation or is there even a common name that is already used
> in the context of cycle index series?
I think that "stretch" is quite ok. Hornegger and Pirastu used "transform",
which doesn't say half as much...
Martin
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