Step variety: Difference between revisions

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For ''r'' ≥ 1, the number of possible patterns (up to rotation) for periodic scales of size ''n'' ≥ ''r'' on ''r'' ordered step sizes ''x''1 > ''x''2 > ... > ''x''''r'' is

<math>\displaystyle{\dfrac{1}{n} \sum_{d\mid n} \phi(d) \sum_{j=1}^r (-1)^{r-j} {r \choose j} j^{n/d},}~~</math>~~

<math>\displaystyle{\dfrac{1}{n} \sum_{d\mid n} \phi(d) \sum_{j=1}^r (-1)^{r-j} {r \choose j} j^{n/d}} \\

~~where <math>~~\~~phi</math> is the Euler totient function. The formula follows from writing the~~ {{~~w|combinatorial species~~}} ~~of [[necklace]]s over ''r'' distinguished letters as the so-called "superposition" <math>~~\~~mathrm~~{~~Bal}^{[r]~~} \~~times \mathcal{C~~}</math> ~~of two species: the species~~ <math>\~~mathrm{Bal}^{[r]}~~</math> ~~of ballots with ''r'' parts (a ''ballot''~~ is ~~a partition with an ordering on~~ the ~~set of parts)~~ and ~~the species~~ <math>~~\mathcal{C}~~</math> of ~~directed cycles, and computing~~ the ~~resulting {{w|generating function}} whose~~ ''n''~~th coefficient is the desired formula.<ref>Bergeron, F., Labelle, G~~.~~, & Leroux, P. (1998). Combinatorial species and tree-like structures (No. 67). Cambridge University Press.</ref>~~

=\displaystyle{\dfrac{1}{n} \sum_{d\mid n} \phi(d) S(n/d, k)}</math>

where <math>\phi</math> is the Euler totient function and <math>S(n, k)</math> is the Stirling number of the second kind for the number of ways to partition an ''n'''element set into ''k'' distinguished parts.

== List of named ternary scales ==

@@ Line 15: / Line 15: @@
 For ''r'' &ge; 1, the number of possible patterns (up to rotation) for periodic scales of size ''n'' &ge; ''r'' on ''r'' ordered step sizes ''x''<sub>1</sub> > ''x''<sub>2</sub> > ... > ''x''<sub>''r''</sub> is
-<math>\displaystyle{\dfrac{1}{n} \sum_{d\mid n} \phi(d) \sum_{j=1}^r (-1)^{r-j} {r \choose j} j^{n/d},}</math>
+<math>\displaystyle{\dfrac{1}{n} \sum_{d\mid n} \phi(d) \sum_{j=1}^r (-1)^{r-j} {r \choose j} j^{n/d}} \\
-where <math>\phi</math> is the Euler totient function. The formula follows from writing the {{w|combinatorial species}} of [[necklace]]s over ''r'' distinguished letters as the so-called "superposition" <math>\mathrm{Bal}^{[r]} \times \mathcal{C}</math> of two species: the species <math>\mathrm{Bal}^{[r]}</math> of ballots with ''r'' parts (a ''ballot'' is a partition with an ordering on the set of parts) and the species <math>\mathcal{C}</math> of directed cycles, and computing the resulting {{w|generating function}} whose ''n''th coefficient is the desired formula.<ref>Bergeron, F., Labelle, G., & Leroux, P. (1998). Combinatorial species and tree-like structures (No. 67). Cambridge University Press.</ref>
+=\displaystyle{\dfrac{1}{n} \sum_{d\mid n} \phi(d) S(n/d, k)}</math>
+where <math>\phi</math> is the Euler totient function and <math>S(n, k)</math> is the Stirling number of the second kind for the number of ways to partition an ''n'''element set into ''k'' distinguished parts.
 == List of named ternary scales ==