Not Another Necklace Post!

Here is how I got into studying necklaces. I started making grids
like this, for hexachords in 12-tET:

7 7 6
8 8 14
8 8 14

The top row is symmetrical hexachords. The second and third rows
represent assymmetrical hexachords and their mirror images. The
rightmost column is hexachords that map into their complements. The
first two columns are Z-related hexachords (always pairs and always
complementary).

Now out of the cells with "14", one set in each ("E"= {0,3,4,5,7,8}
and its inverse, "-E") map directly into their complements under
transposition alone. The 6 in the upper right corner map into
themselves, their complements under both inversion and
transposition. What I have figured out recently is that, there are
always 2^n-1 sets that map into their complements under TnI (where n=
N/2) So there are 32 sets. (It's the right column, leaving off "E"
and its inverse. I just found the formula for inversional-
complementability using Polya.

This grid for all subsets of an 12-tET. The columns are for T(1)-T
(12) in terms of mapping into themselves (or their complements, as
specified).

2^1 2^2 2^3 2^4 2^1 2^6 2^1 2^4 2^3 2^2 2^1 2^12 = 4224 /12 =352 sets
2^6 2^7 2^6 2^7 2^6 2^7 2^6 2^7 2^6 2^7 2^6 2^7 = 1152/12 =96
Symmetrical sets
2^1 2^2 2^3 n/a 2^1 2^6 2^1 n/a 2^3 2^2 2^1 n/a = 96 /12 = 8 Sets
that map into complements under transposition
2^6 n/a 2^6 n/a 2^6 n/a 2^6 n/a 2^6 n/a 2^6 n/a = 384 /12 = 32
Sets that map into complments under TnI

You can pinpoint specific set counts, and do a lot with Polya's
weighting methods to pinpoint specific necklaces. You can go to three
colors (which really complicates the complementability problem!) or
as high as you want. Oh, and you can combine the above rows in
different combinations to get 224 sets reduced for assymmetry, 180
sets reduced for complementability and 122 sets reduced for both.

Recently I found a way to find, for example, a necklace of 6 beads
with 2 beads apiece of 3 colors. There are 16 such necklaces (this is
in Michael Keith's book). Reducing for swapability of colors
(complementability, really) there are exactly (16+14)/6= 5 different
necklaces...

Why am I posting this? So I have a record of my work I guess. I found
a way today to find reduced necklaces of specific bead count, for
example 4,1,1 necklaces reduce to 3 different necklaces.

These formulas hopefully will come in handy some day to work out
metanecklaces for various properties, which might lead to a way of
counting the different interval vectors in an ET. (Number of
different vectors, how many sets use a certain vector, Z-relations,
etc).