On Butler's unimodality result (Q1297764)

Let \(\lambda=(\lambda_i)_{i\in\omega}\) be a ``partition'', i.e., a decreasing sequence of non-negative integers \(\lambda_i\) which are 0 for almost all \(i\). Moreover let \(|\lambda|\) be the sum of these integers. Then \(\lambda\) can represent a finite abelian \(p\)-group \(A\) which is a direct sum of cyclic groups of order \(p^{\lambda_i}\), say \(A\) is of type \(\lambda\). If \(\alpha_\lambda(i,p)\) denotes the number of subgroups of \(A\) of order \(p^i\), then the author shows that the polynomial \(\alpha_\lambda(i,p)-\alpha_\lambda(i-1,p)\) in the variable \(p\) has non-negative coefficients. This was first shown by \textit{L. M. Butler} [Proc. Am. Math. Soc. 101, 771-775 (1987; Zbl 0647.20053)].