Polya's Counting Methods

I am studying the different counting methods of Polya. The first I
learned about on this newsgroup, the second, by way of conversations
with contributors to Sloane's EIS.

I would like to know if the two methods are related mathematically.
The method I am currently studying finds (for example) the 352 set
types in 12-et from the formula 1/n sum (d|n) (phi(n/d)*2^d). The
polynomial method I learned here actually produces the whole
polynomial for C(4)X C(3) with coefficients
1,1,6,19,43,66,80,66,43,19,6,1,1 which add up to 352.

Now, (Method II) uses C for Cyclic, D for Dihedral in the same way as
Method I, but not broken down into 4 X 3 for example. But S(sub 2) is
used to reduce for complementation between black and white beads in a
necklace. Well I know I am just rambling, but since there hasn't been
much activity on this newsgroup for awhile, I thought I would put
this out there. If I could combine the two methods I could generate
(as if we need them) even more lists. I guess a more pointed question
would be to ask if "Polya Method I" could calculate reduction for
complementation (for example, to generate 44 hexachord types, 66
penta/septachord types 43 tetra/octachord types etc. I have a bunch
of tables from a 1961 article in the Illinois Journal of Mathematics
based on "Polya II" and of course all the lists from Gene Ward Smith
based on "Polya I". It would be fun to see if one could combine the
two formulas, (of course they are probably already related at some
level)

Ultimately, I would like to discover mathematics that relates this
musical set theory material with tuning math calculations. Granted
these counting methods can be used to count anything, specifically,
necklaces with black and white beads, it would be fun to relate
these counting methods with considerations about a/tonality . . .

Well Happy New Year!

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@m...> wrote:
>
> I am studying the different counting methods of Polya. The first I
> learned about on this newsgroup, the second, by way of
conversations
> with contributors to Sloane's EIS.
>
> I would like to know if the two methods are related mathematically.
> The method I am currently studying finds (for example) the 352 set
> types in 12-et from the formula 1/n sum (d|n) (phi(n/d)*2^d). The
> polynomial method I learned here actually produces the whole
> polynomial for C(4)X C(3) with coefficients
> 1,1,6,19,43,66,80,66,43,19,6,1,1 which add up to 352.
>
> Now, (Method II) uses C for Cyclic, D for Dihedral in the same way
as
> Method I, but not broken down into 4 X 3 for example. But S(sub 2)
is
> used to reduce for complementation between black and white beads
in a
> necklace. Well I know I am just rambling, but since there hasn't
been
> much activity on this newsgroup for awhile, I thought I would put
> this out there. If I could combine the two methods I could
generate
> (as if we need them) even more lists. I guess a more pointed
question
> would be to ask if "Polya Method I" could calculate reduction for
> complementation (for example, to generate 44 hexachord types, 66
> penta/septachord types 43 tetra/octachord types etc. I have a
bunch
> of tables from a 1961 article in the Illinois Journal of
Mathematics
> based on "Polya II" and of course all the lists from Gene Ward
Smith
> based on "Polya I". It would be fun to see if one could combine
the
> two formulas, (of course they are probably already related at some
> level)
>
> Ultimately, I would like to discover mathematics that relates this
> musical set theory material with tuning math calculations. Granted
> these counting methods can be used to count anything, specifically,
> necklaces with black and white beads, it would be fun to relate
> these counting methods with considerations about a/tonality . . .
>
> Well Happy New Year!

Well, I have been studying the journal article from the 1963
Illinois Journal of Mathematics, which presents some tables that
actually go a little beyond what has been on this newsgroup so far.
In essence, calculating partitions (equivalence between a set and
its complements) is rather difficult mathematics. It is easier to
take the Cyclic Group, and average it with the "kernel of
partitions" or... sets which have a transposition of themselves as a
complement. You can reduce further by taking the "kernel of the
symmetry of partitions" and get down to the 35 hexachord partitions
(reduced for mirror inversion). So, you go from (80 + 8) /2 = 44,
then from (44 + 26) /2 = 35 and get the prized 35 partitions. The
article also has other interesting tables which I will elaborate on
later. Right now I am working with a contributor to Sloane's EIS to
determine exactly how "8" and "26" above are calculated, and I am
getting closer at understanding it intuititvely, at least...

So if this can be calculated with "Polya I" (See above post) then I
would be pretty happy. Any ideas?

Paul Hj

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@m...> wrote:

> Well, I have been studying the journal article from the 1963
> Illinois Journal of Mathematics, which presents some tables that
> actually go a little beyond what has been on this newsgroup so far.

Do you have a better citation?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@c...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@m...> wrote:
>
> > Well, I have been studying the journal article from the 1963
> > Illinois Journal of Mathematics, which presents some tables that
> > actually go a little beyond what has been on this newsgroup so
far.
>
> Do you have a better citation?

Yes. Actually, the article is from the 1961 Illinois Journal of
Mathematics. "Symmetry Types of Periodic Sequences" by E. N. Gilbert
and John Riordan. Recommended by N.J.A. Sloane of Sloane's EIS. I
xeroxed the article from the U of M Library. They present formulas
for both equivalence classes of Cyclic Groups (1/n Sigma_d|n phi(d)q^
(n/d) and for Symmetry Groups (much more involved formula: (1/q!)
Sigma (q!/ksub1!...ksubq!)xsub1^ksub1(xsub2/2)^ksub2...(xsubq)^ksubq
which count complements of a set as equivalent sets. Both the Cyclic
and Symmetry Groups are composed of all the subset content of the
main set (measuring all the equivalence classes). So for n=12 you get
352 for Csubn and 180 for Csubn X Ssub2.

I have also measured DsubnX "M5" (158)and Dsubn X Ssub2 X "M5" (88)
for n=12. I am trying to figure out how to combine the Ssub2 factor
with any of the sets that you presented (Polya polynomials with
permutations). Of course they use the other Polya method, which I am
dubbing "Polya I". "Polya II" is the method I learned about from
Christian Bower of Sloane's EIS (the one above for Cyclic Groups)

Paul