On Butler's unimodality result

For a partition lambda, let alpha(lambda)(i;p) denote the number of subgrou ps of order p(i) in a finite abelian p-group of type lambda. Then alpha(lam bda)(i;p) is a polynomial in p with nonnegative coefficients, which depends only on lambda and i. Butler proved that alpha(lambda)(i;p) - alpha(lambda )(i - 1;p) where 1 less than or equal to i less than or equal to \lambda\/2 has nonnegative coefficients. We prove this fact by using formulas shown b y Stehling.