A relationship between certain colored generalized Frobenius partitions and ordinary partitions
A generalized Frobenius partition, or F-partition, of an integer n is an array \(\begin{pmatrix} a_ 1&a_ 2&...&a_ r \\ b_ 1&b_ 2&...&b_ r \end{pmatrix}\) where the entries in each row are nonnegative integers, \(a_ 1\geq a_ 2\geq...\geq a_ r\), \(b_ 1\geq b_ 2\geq...\geq b_ r\), and \(n=\sum^{r}_{i=1}(a_ i+b_ i+1).\) In this paper the entries in each row are distinct and are taken from k copies of the nonnegative integers distinguished by color. The entries in each row are ordered according to the rule that \(x_ i<y_ j\) if \(x<y\) or if \(x=y\) and \(i<j\), where i,j are integers in the interval [1,k] indicating the colors of x,y. The number of such F-partitions of n with k colors is denoted by \(c\phi_ k(n)\). The number of F-partitions of n with k colors whose order is k under cyclic permutation of the k colors is \[ \overline{c\phi_ k}(n)=\sum_{d| (n,k)}\mu (d)c\phi_{k/d}(n/d). \] It is proved that \(\overline{c\phi_ 5}(n)=5p(5n-1)\) and \(\overline{c\phi_ 7}(n)=7p(7n-2)\) for any positive integer n and, as a corollary, for \(\alpha\) a positive integer, \[ c\phi_ 5(5^{\alpha - 1}n+(\delta_{\alpha}+1)/5)\equiv 0 (mod 5^{\alpha +1})\quad and\quad c\phi_ 7(7^{\alpha -1}n+(\lambda_{\alpha}+2)/7)\equiv 0 (mod 7^{[(\alpha +4)/2]}), \] where \(\delta_{\alpha}\) and \(\lambda_{\alpha}\) are the reciprocals of 24 modulo \(5^{\alpha}\) and \(7^{\alpha}\), respectively. These results are generalizations of a congruence of Ramanujan for ordinary partitions.
scientific article; zbMATH DE number 4112685