Multiple partitions, lattice paths and a Burge-Bressoud-type correspondence

A bijection is presented between (1): partitions with conditions f_j+f_j+1≤k−1 and f₁≤i−1, where f_j is the frequency of the part j in the partition, and (2): sets of k−1 ordered partitions (n⁽¹⁾,n⁽²⁾,…,n^(k−1)) such that and , where m_j is the number of parts in n^(j). This bijection entails an elementary and constructive proof of the Andrews multiple-sum enumerating partitions with frequency conditions. A very natural relation between the k−1 ordered partitions and restricted paths is also presented, which reveals our bijection to be a modification of Bressoud’s version of the Burge correspondence.