Consider an irregular cyclic polygon constructed from a k-sized (1 < k ≤ n) subset of n possible vertices, each corresponding to a point on the unit circle representing one of the nth roots of unity in the complex plane (i.e. an irregular k-gon derived from a regular n-gon). Do such polygons even have a name (cyclic being far too general)? Is there a quick way to check if such a polygon has any symmetries? How might (not)Burnside's Lemma or Pólya enumeration be used to determine and count equivalence classes and congruences?
Or, to put it another way, what might be a generating function for strings S [containing at least one 1] where both (S = H or H0 or H1 or 1G) and (H = 1 or 0H0 or 1H1) and (G = (empty) or 0 or 0G0 or 1G1) from which one might compute B(n,k) [for k=1..n] which enumerates n-lengthed Ss containing exactly k 1s?
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