A correspondence between partitions related to generalizations of the Rogers-Ramanujan identities (Q1147742)

A \(1-1\) correspondence is given between partitions \(p_1+p_2+\ldots+p_n\) of a positive integer \(n\) where \(p_j - p_{j+k-1} \ge 2\) and at most \(i -1\) of the parts equal 1, and the partitions of \(n\) whose hook differences \(d_i\) satisfy \(-i +2\le d_j \le 2k - i - 1\). From Andrews' work these are equinumerous with the number of partitions of \(n\) into parts \(\not\equiv 0, \pm 1\pmod{2k+1}\). The same correspondence is applied to the case of even moduli. Compare \textit{D. M. Bressoud}, J. Number Theory 12, 87--100 (1980; Zbl 0425.10014)].