On the number of equivalence classes arising from partition involutions. II
The authors investigated partition identities while considering classical involutions related with Crank of partitions; and established few new identities relating the number of equivalence classes under the involution and the numbers of partitions satisfying a certain parity condition. Further, authors proved two new fundamental results, as (i). If \(\phi\) be the involution on the set of Frobenius symbol representations of partitions with odd parts distinct, then, for all positive integers \(n\), \(P^{\phi}_{o}(n)=pod_{3}(n)\), where \(pod_{3}(n)\) is the number of partitions with odd parts distinct satisfying that the total number of parts \(\equiv 0 \pmod 3\) is even; and (ii). For a positive integer \(k\), let \(\phi\) be the involution on the set of \(k\)-colored generalized Frobenius partitions, defined by switching the rows, for all positive integers \(n\), \(F^{\phi}_{\overline{C}_{k}, \overline{C}_{k}} (n)=F^{m}_{k}(n)\), where \(F^{m}_{k}(n)\) is the number of \(k\)-colored \(F\)-partitions satisfying that \(m_{F} (\pi) \equiv n + k-1\pmod 2\). The results presented in this article are new, interesting and useful for the researchers. For Part I see [the authors, Int. J. Number Theory 16, No. 5, 925--939 (2020; Zbl 1440.05032)].
scientific article; zbMATH DE number 7358333
scientific article