Problem: How many permutations in S20 have cycle type [13 22 33 41 ]?
Solution: The number of ways to place the integers 1, 2, 3, . . . , 20 into the slots below so
that they fit the cycle type in question is 20!
( )( )( )(
)(
)(
)(
)(
)(
)
However, each permutation σ arises from this procedure in many ways. Looking at each
individual cycle, any member of that cycle can be put in the first slot, and the rest is
determined by σ. Thus for any k cycle, looking within a cycle gives us an overcount by a
factor of k. Thus, the internal arrangement of cycles can be made in 13 · 22 · 33 · 41 ways.
Moreover, the ordering of cycles of the same length does not change the permutation σ.
There are 3 cycles of length one, and hence 3! ways to permute them, 2! ways to permute
cycles of length two, 3! ways to permute cycles of length three, and 1! ways to permute the
cycle of length 4. This overcounts σ by a factor of 3! · 2! · 3! · 1!.
Altogether, the number of permutations in S20 with cycle type [13 22 33 41 ] is
13
·
22
·
33
20!
.
· 41 · 3! · 2! · 3! · 1!
In general, the number of permutations in Sn with cycle type [1a1 2a2 3a3 · · · ] is
n!
.
1a1 2a2 3a3 · · · a1 !a2 !a3 ! · · ·
1