# Unexpected result from CycleIndexPolynomial?

Posted 8 months ago
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 In[2]:= CycleIndexPolynomial[SymmetricGroup[1], {x, x, x}] Out[2]= 1 Shouldn't this be x? I get 1.mathworld .wolfram.com says it is x.
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Posted 8 months ago
 Cycles permutations in WL remove singletons, and therefore the only element of SymmetricGroup[1] is Cycles[{}] instead of Cycles[{{1}}]. This is the source of the mismatch you see.CycleIndexPolynomial takes a third argument to specify the size of the domain of action of the group. Then you can be specific and say In[1]:= CycleIndexPolynomial[SymmetricGroup[1], {x}, 1] Out[1]= x 
Posted 8 months ago
 Thank you Jose. I am now able to include the cycle index for SymmetricGroup[1] in expressing certain integer sequences as a sum of symmetric groups. The calculation below is an example for finding the (N-3)/2 nd term for N = 41 in OEIS A000930In[378]:= AOut[378]= {19, 16, 13, 10, 7, 4, 1}In[384]:= Table[ CycleIndexPolynomial[ SymmetricGroup[A[[k]]], {k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k}, A[[k]]], {k, 1, imax + 1}]Out[384]= {1, 17, 105, 286, 330, 126, 7}In[385]:= Total[%]Out[385]= 872