An extension of Carlitz’s bipartition identity

Abstract:

Carlitz’s bipartition identity is extended to a multipartite partition identity by the introduction of the summatory maximum function: \[ \operatorname {smax}(n_1, n_2, \ldots , n_r) = n_1 + n_2 + \cdots + n_r - (r - 1)\min (n_1, n_2, \ldots , n_r).\] Let $\pi _0(n_1, n_2, \ldots , n_r)$ denote the number of partitions of $({n_1},{n_2}, \ldots ,{n_r})$ in which the minimum coordinate of each part is not less then the summatory maximum of the next part. Let ${\pi _1}({n_1},{n_2}, \ldots ,{n_r})$ denote the number of partitions of $({n_1},{n_2}, \ldots ,{n_r})$ in which each part has one of the $2r - 1$ forms: $(a + 1,a,a, \ldots ,a),(a,a + 1,a, \ldots ,a), \ldots ,(a,a,a, \ldots ,a + 1),(ra + 2,ra + 2, \ldots ,ra + 2),(ra + 3,ra + 3, \ldots ,ra + 3), \ldots ,(ra + r,ra + r, \ldots ,ra + r)$. Theorem: ${\pi _0}({n_1}, \ldots ,{n_r}) = {\pi _1}({n_1}, \ldots ,{n_r})$.