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I want the cycle index of the group of automorphisms of a (say 3 X 3) grid graph. I can produce the elements of the group with:

GroupElements[GraphData[{"Grid", {3, 3}}, "AutomorphismGroup"]]

This gives me: 

{Cycles[{}], Cycles[{{2, 4}, {3, 7}, {6, 8}}], 
 Cycles[{{1, 3}, {4, 6}, {7, 9}}], 
 Cycles[{{1, 3, 9, 7}, {2, 6, 8, 4}}], 
 Cycles[{{1, 7, 9, 3}, {2, 4, 8, 6}}], 
 Cycles[{{1, 7}, {2, 8}, {3, 9}}], Cycles[{{1, 9}, {2, 6}, {4, 8}}], 
 Cycles[{{1, 9}, {2, 8}, {3, 7}, {4, 6}}]}

What I want is something like: $1/8(s_1^9 + 4s_2^3 4s_1^3 + 2s_4^2 s_1 + s_2^4 s_1) $.

The problem seems to be that CycleIndex Needs["Combinatorica"]`.

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    $\begingroup$ Take a look at CycleIndexPolynomial, which is a builtin. $\endgroup$ – Szabolcs Aug 9 '16 at 19:14
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You can use CycleIndexPolynomial like so:

CycleIndexPolynomial[
 GraphAutomorphismGroup@GridGraph[{3, 3}],
 Array[Subscript[s, #] &, 4]
]

$$ \frac{s_1^9}{8}+\frac{1}{2} s_2^3 s_1^3+\frac{1}{8} s_2^4 s_1+\frac{1}{4} s_4^2 s_1 $$

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    $\begingroup$ OK Thank you. I will mention the following as a "possible issue" because it happened to me and caused much aggravation. It is sometimes necessary to specify the "domain of action" (see help section for CycleIndexPolynomial). For example: CycleIndexPolynomial[GraphData[{5, 32}, "AutomorphismGroup"], Table[Subscript[s, i], {i, 1, 8}]] does NOT return the correct polynomial. A 5 must be added as a third argument in CylceIndexPolynomial. $\endgroup$ – Geoffrey Critzer Aug 10 '16 at 16:04

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