Frobenius partitions and the combinatorics of Ramanujan's \(_{1}\psi_{1}\) summation
A partition \(\pi\) of \(n\) is a nonincreasing sequence of natural numbers whose sum is \(n\). Let \(p_{_{A,B}}(n)\) denote the number of generalized Frobenius partitions of \(n\), that is, the number of \(2\)-rowed arrays \[ \left(\begin{matrix} a_1 & a_2 & \cdots & a_m \\ b_1 & b_2 & \cdots & b_m \end{matrix}\right), \] where \(\sum a_i\) is a part of type \(A\) and \(\sum b_i\) is a part of type \(B\) and \[ n=\sum (a_i+b_i+1). \] Let \(C\) denote the ``distinct non-negative parts and unrestricted non-negative overlined parts with \(n>\overline{n}\)''. Let \(f_{r,s}(n)\) be the number of generalized Frobenius partitions counted by \(p_{_{C,C}}(n)\) where there are \(r\) overlined parts in the top row and \(s\) overlined parts in the bottom row. Next, let \(g_{r,s}(n)\) be the number of 4-tuples of partitions \((\pi_1,\pi_2,\pi_3,\pi_4)\) where \(\pi_1\) and \(\pi_4\) are ordinary partitions, \(\pi_2\) and \(\pi_3\) are partitions into distinct parts satisfying \[ \sigma(\pi_1)+\sigma(\pi_2)+\sigma(\pi_3)+\sigma(\pi_4)=n, \] \(r=\mu(\pi_2)+\mu(\pi_3)\) and \(s=\mu(\pi_2)+\mu(\pi_4).\) (Here, \(\sigma(\pi)\) denotes the sum of integers in the sequence \(\pi\) and \(\mu(\pi)\) denotes the number of parts in \(\pi\).) In this article, the authors establish a one to one correspondence between the two sets of partitions of \(n\) described above and show that \(f_{r,s}(n)=g_{r,s}(n)\). They show that their result is equivalent to Ramanujan's \(_1\psi_1\) summation formula. The reviewer wishes to point out that the \(\beta\) given in their example (p. 179) should be \[ \beta = \left(\begin{matrix} \overline{5} & 4 & 3 & \overline{3} & \overline{3} & 2 & 1 & \overline{1} & 0 \\ -4 & -2 & -1 & 5 & \overline{5} & \overline{2}& \overline{2}& \overline{2}& \overline{2}\end{matrix}\right). \]
scientific article; zbMATH DE number 1763800
scientific article