Mathematical construction related to MODMOS (in a way I won't explain yet)

This mathematical construction is puzzling me. Maybe you can help me out, Gene.

Consider the periodic functions Z -> Z of period N. How many such functions have the property that the sum of any sequence of consecutive values of the function is in the set {-1,0,1}? That is, how many integer functions f of period N satisfy

sum_(a<i<b) f(i) in {-1,0,1}

for all pairs of integers (a,b)?

I wrote a little script to tell me the numbers for small values of N, and it spit out the terms of http://oeis.org/A008965 . This just makes me even more puzzled, however, because I can't see a simple mapping from the objects being counted there (necklaces of sets of beads) to the objects I'm counting (functions of period N with a certain property).

For example, here are all the functions I found of period 6:
[1, 0, 0, 0, 0, -1], [0, 1, 0, 0, 0, -1], [1, -1, 1, 0, 0, -1], [0, 0, 1, 0, 0, -1], [1, 0, -1, 1, 0, -1], [0, 1, -1, 1, 0, -1], [1, -1, 0, 1, 0, -1], [0, 0, 0, 1, 0, -1], [1, -1, 1, -1, 1, -1], [0, 0, 1, -1, 1, -1], [0, 1, -1, 0, 1, -1], [0, 0, 0, 0, 1, -1], [0, 0, 0, 0, 0, 0]

and here are the necklaces with 6 total beads:
(2, 3, 1), (2, 1, 1, 1, 1), (2, 2, 2), (2, 4), (3, 3), (4, 1, 1), (1, 1, 1, 1, 1, 1), (3, 1, 1, 1), (2, 2, 1, 1), (1, 5), (2, 1, 3), (6), (2, 1, 2, 1)

I can't see any way to put the functions and the necklaces in 1-to-1 correspondence, but there are 13 of both! And the pattern continues through at least 351!

What's going on here?

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> This mathematical construction is puzzling me. Maybe you can help me out, Gene.
>
> Consider the periodic functions Z -> Z of period N. How many such functions have the property that the sum of any sequence of consecutive values of the function is in the set {-1,0,1}? That is, how many integer functions f of period N satisfy
>
> sum_(a<i<b) f(i) in {-1,0,1}
>
> for all pairs of integers (a,b)?
>
> I wrote a little script to tell me the numbers for small values of N, and it spit out the terms of http://oeis.org/A008965 . This just makes me even more puzzled, however, because I can't see a simple mapping from the objects being counted there (necklaces of sets of beads) to the objects I'm counting (functions of period N with a certain property).
>
> For example, here are all the functions I found of period 6:
> [1, 0, 0, 0, 0, -1], [0, 1, 0, 0, 0, -1], [1, -1, 1, 0, 0, -1], [0, 0, 1, 0, 0, -1], [1, 0, -1, 1, 0, -1], [0, 1, -1, 1, 0, -1], [1, -1, 0, 1, 0, -1], [0, 0, 0, 1, 0, -1], [1, -1, 1, -1, 1, -1], [0, 0, 1, -1, 1, -1], [0, 1, -1, 0, 1, -1], [0, 0, 0, 0, 1, -1], [0, 0, 0, 0, 0, 0]
>
> and here are the necklaces with 6 total beads:
> (2, 3, 1), (2, 1, 1, 1, 1), (2, 2, 2), (2, 4), (3, 3), (4, 1, 1), (1, 1, 1, 1, 1, 1), (3, 1, 1, 1), (2, 2, 1, 1), (1, 5), (2, 1, 3), (6), (2, 1, 2, 1)
>
> I can't see any way to put the functions and the necklaces in 1-to-1 correspondence, but there are 13 of both! And the pattern continues through at least 351!
>
> What's going on here?

I don't know if this helps, but if you consider 1 and -1 as change
and 0 as holding steady, you can consider your sets to be

(5,1) (same as (1,5) necklace)
(4,2) (same as (2,4) necklace)
(1,1,3,1) (same as (3,1,1,1) necklace)
(3,3)
(2,1,2,1)
(1,1,1,2,1) (same as (2,1,1,1,1) necklace)
(1,2,2,1) (same as (2,2,1,1) necklace)
(3,2,1) (same as (2,1,3) necklace)
(1,1,1,1,1,1)
(1,1,1,3) (same as (3,1,1,1), but you have to wrap around, 0,0 at end
(1,2,1,2)
(1,5)
(6)

Well, I couldn't find (2,3,1) and of course you cannot have 0 at the
start...perhaps one of your sets if off, or I am totally wrong here.

PGH

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@> wrote:
> >
> > This mathematical construction is puzzling me. Maybe you can help me out, Gene.
> >
> > Consider the periodic functions Z -> Z of period N. How many such functions have the property that the sum of any sequence of consecutive values of the function is in the set {-1,0,1}? That is, how many integer functions f of period N satisfy
> >
> > sum_(a<i<b) f(i) in {-1,0,1}
> >
> > for all pairs of integers (a,b)?
> >
> > I wrote a little script to tell me the numbers for small values of N, and it spit out the terms of http://oeis.org/A008965 . This just makes me even more puzzled, however, because I can't see a simple mapping from the objects being counted there (necklaces of sets of beads) to the objects I'm counting (functions of period N with a certain property).
> >
> > For example, here are all the functions I found of period 6:
> > [1, 0, 0, 0, 0, -1], [0, 1, 0, 0, 0, -1], [1, -1, 1, 0, 0, -1], [0, 0, 1, 0, 0, -1], [1, 0, -1, 1, 0, -1], [0, 1, -1, 1, 0, -1], [1, -1, 0, 1, 0, -1], [0, 0, 0, 1, 0, -1], [1, -1, 1, -1, 1, -1], [0, 0, 1, -1, 1, -1], [0, 1, -1, 0, 1, -1], [0, 0, 0, 0, 1, -1], [0, 0, 0, 0, 0, 0]
> >
> > and here are the necklaces with 6 total beads:
> > (2, 3, 1), (2, 1, 1, 1, 1), (2, 2, 2), (2, 4), (3, 3), (4, 1, 1), (1, 1, 1, 1, 1, 1), (3, 1, 1, 1), (2, 2, 1, 1), (1, 5), (2, 1, 3), (6), (2, 1, 2, 1)
> >
> > I can't see any way to put the functions and the necklaces in 1-to-1 correspondence, but there are 13 of both! And the pattern continues through at least 351!
> >
> > What's going on here?
>
> I don't know if this helps, but if you consider 1 and -1 as change
> and 0 as holding steady, you can consider your sets to be
>
> (5,1) (same as (1,5) necklace)
> (4,2) (same as (2,4) necklace)
> (1,1,3,1) (same as (3,1,1,1) necklace)
> (3,3)
> (2,1,2,1)
> (1,1,1,2,1) (same as (2,1,1,1,1) necklace)
> (1,2,2,1) (same as (2,2,1,1) necklace)
> (3,2,1) (same as (2,1,3) necklace)
> (1,1,1,1,1,1)
> (1,1,1,3) (same as (3,1,1,1), but you have to wrap around, 0,0 at end
> (1,2,1,2)
> (1,5)
> (6)
>
> Well, I couldn't find (2,3,1) and of course you cannot have 0 at the
> start...perhaps one of your sets if off, or I am totally wrong here.
>
> PGH

Should have waited until Gene answered. However, there might be
some correspondence. I redid my sets and get

51, 24, 3111, 33, 2121, 2112, 1221, 42, 111111, 3111, 2121, 51, 6.

So 5 necklace types are used twice, and 3 one time. So 5 are not
used in your sets. The "pure" ones are used once, (33, 111111, 6)
and half of the remainder are used twice. There might be an algorithm
that could explain this all the way up to 351 (which is also
the number of total necklaces in 12-tET, btw). Ones not used are
21111, 222, 231, 213, and 411. It looks like the ones with an
even number of digits are used, and those with an odd are not, except for 6. Perhaps one could say odd primes lengths are not used here...

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:

> Should have waited until Gene answered.

Why? I didn't see the answer straight off the bat, so I am hoping you will figure it out, as I'm thinking about subgroups at the moment.

On Mon, Mar 28, 2011 at 5:15 AM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>
> For example, here are all the functions I found of period 6:
> [1, 0, 0, 0, 0, -1], [0, 1, 0, 0, 0, -1], [1, -1, 1, 0, 0, -1], [0, 0, 1, 0, 0, -1], [1, 0, -1, 1, 0, -1], [0, 1, -1, 1, 0, -1], [1, -1, 0, 1, 0, -1], [0, 0, 0, 1, 0, -1], [1, -1, 1, -1, 1, -1], [0, 0, 1, -1, 1, -1], [0, 1, -1, 0, 1, -1], [0, 0, 0, 0, 1, -1], [0, 0, 0, 0, 0, 0]
>
> and here are the necklaces with 6 total beads:
> (2, 3, 1), (2, 1, 1, 1, 1), (2, 2, 2), (2, 4), (3, 3), (4, 1, 1), (1, 1, 1, 1, 1, 1), (3, 1, 1, 1), (2, 2, 1, 1), (1, 5), (2, 1, 3), (6), (2, 1, 2, 1)
>
> I can't see any way to put the functions and the necklaces in 1-to-1 correspondence, but there are 13 of both! And the pattern continues through at least 351!

Keenan, can you explain the latter notation? How does one interpret (2,3,1)?

-Mike

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@> wrote:
> >
> >
> >
> > --- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@> wrote:
> > >
> > > This mathematical construction is puzzling me. Maybe you can help me out, Gene.
> > >
> > > Consider the periodic functions Z -> Z of period N. How many such functions have the property that the sum of any sequence of consecutive values of the function is in the set {-1,0,1}? That is, how many integer functions f of period N satisfy
> > >
> > > sum_(a<i<b) f(i) in {-1,0,1}
> > >
> > > for all pairs of integers (a,b)?
> > >
> > > I wrote a little script to tell me the numbers for small values of N, and it spit out the terms of http://oeis.org/A008965 . This just makes me even more puzzled, however, because I can't see a simple mapping from the objects being counted there (necklaces of sets of beads) to the objects I'm counting (functions of period N with a certain property).
> > >
> > > For example, here are all the functions I found of period 6:
> > > [1, 0, 0, 0, 0, -1], [0, 1, 0, 0, 0, -1], [1, -1, 1, 0, 0, -1], [0, 0, 1, 0, 0, -1], [1, 0, -1, 1, 0, -1], [0, 1, -1, 1, 0, -1], [1, -1, 0, 1, 0, -1], [0, 0, 0, 1, 0, -1], [1, -1, 1, -1, 1, -1], [0, 0, 1, -1, 1, -1], [0, 1, -1, 0, 1, -1], [0, 0, 0, 0, 1, -1], [0, 0, 0, 0, 0, 0]
> > >
> > > and here are the necklaces with 6 total beads:
> > > (2, 3, 1), (2, 1, 1, 1, 1), (2, 2, 2), (2, 4), (3, 3), (4, 1, 1), (1, 1, 1, 1, 1, 1), (3, 1, 1, 1), (2, 2, 1, 1), (1, 5), (2, 1, 3), (6), (2, 1, 2, 1)
> > >
> > > I can't see any way to put the functions and the necklaces in 1-to-1 correspondence, but there are 13 of both! And the pattern continues through at least 351!
> > >
> > > What's going on here?
> >
> > I don't know if this helps, but if you consider 1 and -1 as change
> > and 0 as holding steady, you can consider your sets to be
> >
> > (5,1) (same as (1,5) necklace)
> > (4,2) (same as (2,4) necklace)
> > (1,1,3,1) (same as (3,1,1,1) necklace)
> > (3,3)
> > (2,1,2,1)
> > (1,1,1,2,1) (same as (2,1,1,1,1) necklace)
> > (1,2,2,1) (same as (2,2,1,1) necklace)
> > (3,2,1) (same as (2,1,3) necklace)
> > (1,1,1,1,1,1)
> > (1,1,1,3) (same as (3,1,1,1), but you have to wrap around, 0,0 at end
> > (1,2,1,2)
> > (1,5)
> > (6)
> >
> > Well, I couldn't find (2,3,1) and of course you cannot have 0 at the
> > start...perhaps one of your sets if off, or I am totally wrong here.
> >
> > PGH
>
> Should have waited until Gene answered. However, there might be
> some correspondence. I redid my sets and get
>
> 51, 24, 3111, 33, 2121, 2112, 1221, 42, 111111, 3111, 2121, 51, 6.
>
> So 5 necklace types are used twice, and 3 one time. So 5 are not
> used in your sets. The "pure" ones are used once, (33, 111111, 6)
> and half of the remainder are used twice. There might be an algorithm
> that could explain this all the way up to 351 (which is also
> the number of total necklaces in 12-tET, btw). Ones not used are
> 21111, 222, 231, 213, and 411. It looks like the ones with an
> even number of digits are used, and those with an odd are not, except for 6. Perhaps one could say odd primes lengths are not used here...
>

Actually, 6 could be interpreted as having 0 length, if one counts breakpoints instead
of segments. Then all the "good" necklaces have even length. Mike, I won't tromp
on your question to Keenan but those numbers (2,3,1) are just partitions of 6.
In this case, you could have 2 red, 3 white and 1 blue bead for example. ) Even ones
only need two colors, such as black and white, which are the ones used in conjunction
with the functions up above.

PGH

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@> wrote:
> >
> >
> >
> > --- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@> wrote:
> > >
> > >
> > >
> > > --- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@> wrote:
> > > >
> > > > This mathematical construction is puzzling me. Maybe you can help me out, Gene.
> > > >
> > > > Consider the periodic functions Z -> Z of period N. How many such functions have the property that the sum of any sequence of consecutive values of the function is in the set {-1,0,1}? That is, how many integer functions f of period N satisfy
> > > >
> > > > sum_(a<i<b) f(i) in {-1,0,1}
> > > >
> > > > for all pairs of integers (a,b)?
> > > >
> > > > I wrote a little script to tell me the numbers for small values of N, and it spit out the terms of http://oeis.org/A008965 . This just makes me even more puzzled, however, because I can't see a simple mapping from the objects being counted there (necklaces of sets of beads) to the objects I'm counting (functions of period N with a certain property).
> > > >
> > > > For example, here are all the functions I found of period 6:
> > > > [1, 0, 0, 0, 0, -1], [0, 1, 0, 0, 0, -1], [1, -1, 1, 0, 0, -1], [0, 0, 1, 0, 0, -1], [1, 0, -1, 1, 0, -1], [0, 1, -1, 1, 0, -1], [1, -1, 0, 1, 0, -1], [0, 0, 0, 1, 0, -1], [1, -1, 1, -1, 1, -1], [0, 0, 1, -1, 1, -1], [0, 1, -1, 0, 1, -1], [0, 0, 0, 0, 1, -1], [0, 0, 0, 0, 0, 0]
> > > >
> > > > and here are the necklaces with 6 total beads:
> > > > (2, 3, 1), (2, 1, 1, 1, 1), (2, 2, 2), (2, 4), (3, 3), (4, 1, 1), (1, 1, 1, 1, 1, 1), (3, 1, 1, 1), (2, 2, 1, 1), (1, 5), (2, 1, 3), (6), (2, 1, 2, 1)
> > > >
> > > > I can't see any way to put the functions and the necklaces in 1-to-1 correspondence, but there are 13 of both! And the pattern continues through at least 351!
> > > >
> > > > What's going on here?
> > >
> > > I don't know if this helps, but if you consider 1 and -1 as change
> > > and 0 as holding steady, you can consider your sets to be
> > >
> > > (5,1) (same as (1,5) necklace)
> > > (4,2) (same as (2,4) necklace)
> > > (1,1,3,1) (same as (3,1,1,1) necklace)
> > > (3,3)
> > > (2,1,2,1)
> > > (1,1,1,2,1) (same as (2,1,1,1,1) necklace)
> > > (1,2,2,1) (same as (2,2,1,1) necklace)
> > > (3,2,1) (same as (2,1,3) necklace)
> > > (1,1,1,1,1,1)
> > > (1,1,1,3) (same as (3,1,1,1), but you have to wrap around, 0,0 at end
> > > (1,2,1,2)
> > > (1,5)
> > > (6)
> > >
> > > Well, I couldn't find (2,3,1) and of course you cannot have 0 at the
> > > start...perhaps one of your sets if off, or I am totally wrong here.
> > >
> > > PGH
> >
> > Should have waited until Gene answered. However, there might be
> > some correspondence. I redid my sets and get
> >
> > 51, 24, 3111, 33, 2121, 2112, 1221, 42, 111111, 3111, 2121, 51, 6.
> >
> > So 5 necklace types are used twice, and 3 one time. So 5 are not
> > used in your sets. The "pure" ones are used once, (33, 111111, 6)
> > and half of the remainder are used twice. There might be an algorithm
> > that could explain this all the way up to 351 (which is also
> > the number of total necklaces in 12-tET, btw). Ones not used are
> > 21111, 222, 231, 213, and 411. It looks like the ones with an
> > even number of digits are used, and those with an odd are not, except for 6. Perhaps one could say odd primes lengths are not used here...
> >
>
> Actually, 6 could be interpreted as having 0 length, if one counts breakpoints instead
> of segments. Then all the "good" necklaces have even length. Mike, I won't tromp
> on your question to Keenan but those numbers (2,3,1) are just partitions of 6.
> In this case, you could have 2 red, 3 white and 1 blue bead for example. ) Even ones
> only need two colors, such as black and white, which are the ones used in conjunction
> with the functions up above.
>
> PGH

Just one more thing -- of course those five necklaces are used twice
because of the swapping of colors. If one makes +1 white and -1
black beads (plus the zeros that follow each one, respectively)
you will need them twice, because of the signs. Let me see ---

6 has zero length (wrap around of 0,0,0,0,0,0 string)
51, 42, and 33 have 2-length
222, 231, 213, 411 have 3-length
3111, 2121, 1122 have 4-length
21111 have 5-length
111111 have 6-length

This is 1, 0, 3, 4, 3, 1, 1, where evens are 8 and odds are 5
33, 111111, 6 map into themselves under color-complementability.

If I could see other function chains with their corresponding
necklaces I think I could find the formula. However, it is only
a correspondence of cardinality, not a one-to-one matching, I am
sure. Remember you can only use even necklaces to match your
functions and some are used twice. Hope I don't sound condescending
that is not how I feel....I just love necklace math!

PGH

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> Keenan, can you explain the latter notation? How does one interpret (2,3,1)?
>
> -Mike

(2,3,1) is a necklace with 6 beads on it, in groups of 2, 3, and 1, in that order. So (1,2,3) would be the same necklace (since it's just rotated around), but (1,3,2) would be a different necklace (the mirror image of the first, which is not considered equivalent).

It's just the definition of this OEIS sequence: http://oeis.org/A008965

Paul, your attempt at making a 1-to-1 correspondence is a good shot, and practically the same as what I was thinking, but it doesn't work and the correct correspondence uses a slightly different basic idea.

I asked this question at Math Stack Exchange and somebody gave the correct answer surprisingly quickly: http://math.stackexchange.com/questions/29471/number-of-periodic-integer-functions-with-a-certain-property

Since you asked for more examples, here are all the examples up to n=7 in an abbreviated notation (which should be obvious, but please tell me if it's not):

0 <---> 1

00 <---> 11
+- <---> 2

000 <---> 111
0+- <---> 12
+0- <---> 3

0000 <---> 1111
00+- <---> 112
0+0- <---> 13
+00- <---> 4
+-+- <---> 22

00000 <---> 11111
000+- <---> 1112
00+0- <---> 113
0+00- <---> 14
0+-+- <---> 122
+000- <---> 5
+0-+- <---> 32

000000 <---> 111111
0000+- <---> 11112
000+0- <---> 1113
00+00- <---> 114
00+-+- <---> 1122
0+000- <---> 15
0+0-+- <---> 132
0+-0+- <---> 1212
0+-+0- <---> 123
+0000- <---> 6
+00-+- <---> 42
+0-+0- <---> 33
+-+-+- <---> 222

0000000 <---> 1111111
00000+- <---> 111112
0000+0- <---> 11113
000+00- <---> 1114
000+-+- <---> 11122
00+000- <---> 115
00+0-+- <---> 1132
00+-0+- <---> 11212
00+-+0- <---> 1123
0+0000- <---> 16
0+00-+- <---> 142
0+0-0+- <---> 1312
0+0-+0- <---> 133
0+-+00- <---> 124
0+-+-+- <---> 1222
+00000- <---> 7
+000-+- <---> 52
+00-+0- <---> 43
+0-+-+- <---> 322

Keenan

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
>
>
> Paul, your attempt at making a 1-to-1 correspondence is a good shot, and practically the same as what I was thinking, but it doesn't work and the correct correspondence uses a slightly different basic idea.
>
> I asked this question at Math Stack Exchange and somebody gave the correct answer surprisingly quickly: http://math.stackexchange.com/questions/29471/number-of-periodic-integer-functions-with-a-certain-property
>
> Since you asked for more examples, here are all the examples up to n=7 in an abbreviated notation (which should be obvious, but please tell me if it's not):
>
> 0 <---> 1
>
> 00 <---> 11
> +- <---> 2
>
> 000 <---> 111
> 0+- <---> 12
> +0- <---> 3
>
> 0000 <---> 1111
> 00+- <---> 112
> 0+0- <---> 13
> +00- <---> 4
> +-+- <---> 22
>
> 00000 <---> 11111
> 000+- <---> 1112
> 00+0- <---> 113
> 0+00- <---> 14
> 0+-+- <---> 122
> +000- <---> 5
> +0-+- <---> 32
>
> 000000 <---> 111111
> 0000+- <---> 11112
> 000+0- <---> 1113
> 00+00- <---> 114
> 00+-+- <---> 1122
> 0+000- <---> 15
> 0+0-+- <---> 132
> 0+-0+- <---> 1212
> 0+-+0- <---> 123
> +0000- <---> 6
> +00-+- <---> 42
> +0-+0- <---> 33
> +-+-+- <---> 222
>
> 0000000 <---> 1111111
> 00000+- <---> 111112
> 0000+0- <---> 11113
> 000+00- <---> 1114
> 000+-+- <---> 11122
> 00+000- <---> 115
> 00+0-+- <---> 1132
> 00+-0+- <---> 11212
> 00+-+0- <---> 1123
> 0+0000- <---> 16
> 0+00-+- <---> 142
> 0+0-0+- <---> 1312
> 0+0-+0- <---> 133
> 0+-+00- <---> 124
> 0+-+-+- <---> 1222
> +00000- <---> 7
> +000-+- <---> 52
> +00-+0- <---> 43
> +0-+-+- <---> 322
>
> Keenan
>

[Thanks --- but my whole point was [now moot] that it is NOT a one-to-one correspondence, it is an equivalence of the cardinalities of the
two things (if you looked at my final message).]

I see here that the mapping is based on a different idea, where + and - frame the zeros, interesting...it's all a little like Fourier Analysis, isn't it? So here, we do have a one-to-one correspondence (obviously better). I'll try Math Stack Exchange, I have been posting on Math Overflow, also excellent. This necklace theory pops up all over, with Polya math, and even with Modular Forms...It would be
fun now to apply the Polya equations to your chains, and see if
they shed any more light on things, I used to know it for the 352
necklaces (this includes the null set) which would be easily lookupable.

Thanks. PGH

On Tue, Mar 29, 2011 at 3:04 PM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > Keenan, can you explain the latter notation? How does one interpret (2,3,1)?
> >
> > -Mike
>
> (2,3,1) is a necklace with 6 beads on it, in groups of 2, 3, and 1, in that order. So (1,2,3) would be the same necklace (since it's just rotated around), but (1,3,2) would be a different necklace (the mirror image of the first, which is not considered equivalent).
>
> It's just the definition of this OEIS sequence: http://oeis.org/A008965

So if the beads are white, black, and red, then (2,3,1) would be
something like wwbbbr? What would wwbrrb be, (2,1,2,1)? And so it'd be
the same as wwbwwb, which is also (2,1,2,1)?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> So if the beads are white, black, and red, then (2,3,1) would be
> something like wwbbbr? What would wwbrrb be, (2,1,2,1)? And so it'd be
> the same as wwbwwb, which is also (2,1,2,1)?
>
> -Mike

That's right, but if you want a way to visualize it that makes more sense, don't think about different colored beads. Instead, think of necklaces with groups of identical beads, and separators such as knots.

So (2,3,1) is (2 beads, knot, 3 beads, knot, 1 bead, knot).

Keenan

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
>
>
> Paul, your attempt at making a 1-to-1 correspondence is a good shot, and practically the same as what I was thinking, but it doesn't work and the correct correspondence uses a slightly different basic idea.
>
> I asked this question at Math Stack Exchange and somebody gave the correct answer surprisingly quickly: http://math.stackexchange.com/questions/29471/number-of-periodic-integer-functions-with-a-certain-property

This would be a good comment to send to OEIS.