Step variety: Difference between revisions

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<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}} 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 ~~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>

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 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>

== List of named ternary scales ==

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 <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}} 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 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>
+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 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>
 == List of named ternary scales ==