Numbers of conjugate classes of symmetric and alternating groups (Q1082432)

Let d(n) be the excess of the number of even conjugate classes of \(S_ n\) over that of odd conjugate classes of \(S_ n\), and q(n) the number of splitting classes of \(S_ n\). In this paper a recurrence formula for d(n) and one for q(n) are given. As a recurrence formula for the number p(n) of conjugate classes of \(S_ n\) is known, one can make use of p(n), d(n) and q(n) to calculate the numbers of even and odd conjugate classes of \(S_ n\) and that of conjugate classes of \(A_ n\). By use of the graphical representation of partitions, the author proves the identity \(d(n)=q(n)\) when \(n\geq 2\), which seems to have been mentioned first by Sylvester. It follows from this identity that the number of even conjugate classes of \(S_ n\) \((n>2)\) is always greater than the number of odd conjugate classes. Finally, as a consequence of this identity, the author proves again another of Sylvester's theorems.