In order to gain some insight into this problem, define a random variable to be the number of eigenvalues of equal to . The variable is already well understood; see, for example, [1], [2]. Presented here is a brief explanation of what happens to the mean of as .

First consider the case when .
Every cycle in a permutation
produces the
eigenvalue 1, so the number of eigenvalues at
will equal the total number of
cycles in .
Recall that the number of cycles in
is the sum of all the
values of *C*_{k} in the cycle structure. Thus,

(14) |

and the mean of

(15) |

For large

(16) |

Similar reasoning can be used to see that in general, if
with *p* and *q*relatively prime, then

(17) |

If is an irrational multiple of , then no eigenvalues can occur there, so for all