Algorithmically List Representatives of Conjugacy Classes of Symmetric Group

Question

Are there any pre-written algorithms for listing representatives for each cycle type for $S_n$, the group of permutations of size $n$? Preferably, they would be listed with for example in $S_4$ the identity = [1,2,3,4], (12) = [2,1,3,4] etc. If not, does anyone have any idea on how an algorithm for such a purpose would work? I cannot quite figure one out nor find one through searching.

Answer 1

There are well known algorithms to list all the partitions of an integer. Here is a Python reference.

For each instance, for example, $ 7 = 3 +2 + 1 +1$, write the cycle structure $$ (xxx)(xx)(x)(x), $$ fill the cycles with the integers in order $$ (123)(45)(6)(7), $$ and loop to find the conjugacy class representative

1 2 3 4 5 6 7
2 3 1 5 4 6 7

in the form you want.