A little puzzle

Hello again. I'm sure by now the endless talk about mathematical necklaces has left you gagging on the words Polya's Enumeration Theorem, but bear with me.

Consider a 12-bead necklace constructed of black and white beads, any number of each, so long as it adds to 12 exactly. No, this necklace cannot be taken off and put on the other way; you have to stick your head in the same way every time, otherwise

How many such necklaces are there? As is intuitively obvious even to the most casual observer, 352. But given a specific combination of black and white beads, how many can be constructed? Well, twelve white beads is easy: there's only one option, and it's the same with twelve black beads, and similarly there is only one way to arrange a single bead of one color along with eleven of another color. For counting (though not aesthetic) purposes, black and white are interchangeable: for instance, you'll get the same count if you're using ten black and two white as you will with ten white and two black. Take the latter option: how many necklaces can you form with two black beads and ten white? Well, the two black beads could be right next to each other, or one bead removed, two beads removed, and so on until they have five white beads between them on either side. This yields six necklaces of two black beads and ten white, and six more of the opposite arrangement.

The other arrangements are harder to count by inspection, but with the help of a visualization tool I found once, I cheated straight to the answer of how the 352 necklaces break out into the 13 possible combinations of white and black beads (see below). The first two ones represent using all beads of one color, and using eleven beads of one color and one of the other. The two ones at the end represent using just one bead of the one color, and using all beads of the other color (Make sense? Of course not. It's 10:30 PM and I was up at 4:20 this morning) In the case of using six and six, there are eighty possibilities:

1 + 1 + 6 + 19 + 43 + 66 + 80 + 66 + 43 + 19 + 6 + 1 + 1 = 352

What the heck does any of this have to do with music theory? Here's a hint: https://www.google.com/search?tbo=p&tbm=bks&q=isbn:1619114860

My apologies for being pedantic and cocky.