neclaces are elegant, but bracelets look gross: consider the table of bracelet polynomials, n beads, c<=n colours, substitute colour k by Exp[2 Pi I k/c], Table[NecklacePolynomial[n,Exp[2 I Pi Range[c]/c],Dihedral]//FullSimplify,{n,13},{c,n}] and find: {1}, {1,1}, {1,0,1}, {1,2,0,1}, {1,0,0,0,2}, {1,3,2,0,0,1}, {1,0,0,0,0,0,3}, {1,6,0,3,0,0,0,2}, {1,0,4,0,0,0,0,0,3}, {1,10,0,0,6,0,0,0,0,2}, {1,0,0,0,0,0,0,0,0,0,5}, {1,20,8,6,0,4,0,0,0,0,0,2}, {1,0,0,0,0,0,0,0,0,0,0,0,6}, and that might very well be: if c|n then if(c=1,2,1)*(1/2/c)* (if(c=2,2^(n/c),0)+ Sum(d|(n/c), phi(c*d)*c^(n/c/d)/(n/c)) else 0 or, Table[If[MemberQ[Divisors[n],c],If[c===1,2,1]*(1/2/c) *(If[c===2,2^(n/c),0]+Fold[#1+ EulerPhi[c #2]*c^(n/#2/c) /(n/c)&,0,Divisors[n/c]]),0],{n,13},{c,n}] which is not nice at all, with ugly If's showing definite lack of elegance (?!) Anyone to de-grossify this? But I honestly admit still not to grasp (=grok) the 'why' of these triviae. So, I give up. Wouter.