Polya Powers and Polynomials

Lately, I have been familiarizing myself with Polya's methods. I
finally see the relationship between the powers of 2 method and the
polynomial one. (Polynomials are just expansions of powers). I've
managed to integrate everything into the polynomial method, with a
little fudging, and also see how the formulas on EIS are derived from
these methods. (A003239, A005648, A045629, A006840, A000031, A000013,
etc). With hexachords in 12t-ET, you get 35 sets, reducing for
reversibility and complementability. This is the same as reducing for Z-
relations, since all Z-related sets are complementable in C(12,6).
Very helpful is a paper from 1961 called "Symmetry Types of Periodic
Sequences"

Very striking is the fact that there are 26 hexachords which are
symmetrical OR reversible complementable (or both). It's (C(6,3)-x)/2 +
x + (2^5-x)/2. You don't need to know x! (It's 6, sets which are both
complementable and symmetrical). That's Sloane's A045674. Adding 26
back into the chain from 0-12 gives 64 "symmetrical partitions" total,
remembering that a (hexachord) partition is symmetrical if it's
complement is reversible-complementable.