Identities for self-conjugate 7- and 9-core partitions (Q2880329)

In a Ferrers-Young diagram of a self-conjugate partition \(\lambda\) of \(n\), a natural number, where \(\lambda_i\) is the number of nodes in the \(i\)th row and \(\lambda_j\) in the \(j\)th column, the Hook number \(H(i, j)\) of the \((i, j)\) node is \(\lambda_i+ \lambda_j= i- j+ 1\). A \(t\)-core partition is a partition with no hook number divisible by \(t\). If \(\text{asc}_t(n)\) is the number of self-conjugate \(t\)-core partitions of \(n\), one can find generating functions for this \(\text{asc}_t(n)\) for \(t\) both even and odd.NEWLINENEWLINE The authors introduce a similar function \(\text{add}_t(n)\) for ``double distinct'' partitions of \(2n\) into distinct summands and derive similar generating functions, using Ramanujan's theta functions. Specifically we see that \(\text{asc}_7(8n+ 7) = 0\), \(\text{asc}_9(8n+ 10) = \text{asc}_9(2n)\), \(\text{asc}_3(4n) = \text{add}_3(n)\), \(\text{add}_3(2P_k) = 1\), where \(P_k\) are the generalized pentagonal numbers \(k(3k \pm 1)/2\), and \(\text{asc}_5(2n) = \text{add}_5(n)\). Some of the computations are formidable.