Imagine a necklace with n beads of up to n colours, say n=3 beads made of colours x[1], x[2] and x[3], then these are counted by: Table[NecklacePolynomial[n,Array[x,n],Cyclic],{n,3,3}] {x[1]^3 + x[1]^2*x[2] + x[1]*x[2]^2 + x[2]^3 + x[1]^2*x[3] + 2*x[1]*x[2]*x[3] + x[2]^2*x[3] + x[1]*x[3]^2 + x[2]*x[3]^2 + x[3]^3} now, change the colours into complex roots of 1 x[k] -> E^(2I Pi k/n) so that they do a 'complex cancellation' on the unit circle, and, hey presto, the whole caboodle collapses to EulerPhi[n]: EulerPhi[n] == NecklacePolynomial[n,E^(2I Pi Range[n]/n),Cyclic] which is nice... Wouter.