Group Permanents of Abelian $p$-Groups and Young Diagrams

We study the number $\mathrm{N}(\text{Per}(G_{\lambda }))$ of distinct monomials with nonzero coefficients in the group permanent of an abelian $p$-group $G_{\lambda }$ associated with a partition $\lambda$ of a positive integer $N$. First, we derive an explicit formula for $\mathrm{N}(\text{Per}(G_{\lambda }))$ in terms of the partial column sums of the Young diagram of $\lambda$. Next, we show that the relative order of the values $\mathrm{N}(\text{Per}(G_{\lambda }))$ is determined by a lexicographic comparison of the conjugate Young diagrams. Finally, we investigate congruence properties of $\mathrm{N}(\text{Per}(G_{\lambda }))$ for abelian $p$-groups and establish a criterion involving Wolstenholme primes.