Unexpected result from CycleIndexPolynomial?

Questions
1

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.

2

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
3

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 A000930

In[378]:= A

Out[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

4

Great! I guess it would be possible to use in general Table[k, Max[A]] instead of the list {k, k, …, k}.

5

Yes, that helps streamline the command, particularly for large Max[A] . Thanks!