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'' over ''r'' ordered step sizes ''x''1 > ''x''2 > ... > ''x''''r'' is

<math>\displaystyle{\dfrac{1}{n} \sum_{~~km =~~ n~~\\k,m\geq 1~~} \phi(k) \sum_{j=1}^r (-1)^{r-j} {r \choose j} j^m,}</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},}</math>

where <math>\phi</math> is the Euler totient function. The formula follows from writing the {{w|combinatorial species}} (finite structure) of [[necklace]]s over ''r'' 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 whose parts are ordered) and the species <math>\mathcal{C}</math> of ordered 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>

@@ Line 12: / Line 12: @@
 For ''r'' &ge; 1, the number of possible patterns (up to rotation) for periodic scales of size ''n'' &ge; ''r'' over ''r'' ordered step sizes ''x''<sub>1</sub> > ''x''<sub>2</sub> > ... > ''x''<sub>''r''</sub> is
-<math>\displaystyle{\dfrac{1}{n} \sum_{km = n\\k,m\geq 1} \phi(k) \sum_{j=1}^r (-1)^{r-j} {r \choose j} j^m,}</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},}</math>
 where <math>\phi</math> is the Euler totient function. The formula follows from writing the {{w|combinatorial species}} (finite structure) of [[necklace]]s over ''r'' 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 whose parts are ordered) and the species <math>\mathcal{C}</math> of ordered 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>