On the Pólya Enumeration Theorem

Simple formulas for the number of different cyclic and dihedral necklaces containing nj beads of the j-th color, j ≤ m and ∑m j=1 nj = N , are derived. Among a vast number of counting problems one of the most popular is a necklace enumeration. A cyclic necklace is a coloring in m colors of the vertices of a regular N–gon, where two colorings are equivalent if one can be obtained from the other by a cyclic symmetry CN , e.g. colored beads are placed on a circle, and the circle may be rotated (without reflections). A basic enumeration problem is then: for given m and N = ∑m j=1 nj, how many different cyclic necklaces containing nj beads of the j-th color are there. The answer follows by an application of the Pólya’s theorem [1]: the number γ(CN ,n m) of different cyclic necklaces is the coefficient of x1 1 · . . . · x nm m in the cycle index ZCN (xi) = 1 N ∑ g|N φ(g)X g , Xg = x g 1 + . . .+ x g m , (1) where φ(g) denotes the Euler totient function and nm denotes a tuple (n1, . . . , nm). In this article we prove that γ (CN ,n ) = 1 N ∑ d|∆ φ(d)P (k) , where P (k) = (k1 + . . .+ km)! ∏m j=1 kj! , kj = nj d , (2) and ∆ denotes a great common divisor gcdnm of the tuple nm. We denote also km = (k1, . . . , km). Note that the term x1 1 · . . . · x nm m does appear only once in the multinomial series expansion (MSE) of (1) with a weight P (nm) when g = 1, X 1 −→ P (n ) x1 1 · . . . · x nm m , where N = n1 + . . . + nm . (3) Show that for g > 1 the polynomial ZCN (xi) contributes in γ (CN ,n m) if and only if ∆ > 1. We prove that if g|N and g 6 |∆ then the term x1 1 · . . . · x nm m does not appear in MSE of (1). 1 Denote N/g = L, 1 < L < N and consider MSE of (1) X g = l1+···+lm=L ∑ li≥0 P (l) x1 1 · . . . · x glm m , (4) where lm denotes a tuple (l1, . . . , lm). However MSE in (4) does not contribute in γ (CN ,n m) since g 6 |∆, i.e. we cannot provide such g that gli = ni holds for all i = 1, . . . ,m. Thus, we have reduced expression (1) by summing only over the divisors d of ∆, ZCN (xi) = 1 N ∑ d|∆ φ(d)X N/d d . (5) Denoting kj = nj/d, N/d = K = k1 + . . . + km, and considering MSE of (5) we obtain X d −→ P (k ) x1 1 · . . . · x dkm m = P (k ) x1 1 · . . . · x nm m . (6) Combining (5) and (6) we arrive at (2). It is easy to extend the explicit formula (2) to the case of dihedral necklaces where two colorings are equivalent if one can be obtained from the other by a dihedral symmetry DN , e.g. colored beads are placed on a circle, and the circle may be rotated and reflected. Start with the cycle indices [1] 2ZDN (xi) = ZCN (xi) + 