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

A bijection is presented between (1): partitions with conditions fj+fj+1≤k−1fj+fj+1≤k−1 and f1≤i−1f1≤i−1, where fjfj is the frequency of the part jj in the partition, and (2): sets of k−1k−1 ordered partitions (n(1),n(2),…,n(k−1))(n(1),n(2),…,n(k−1)) such that nℓ(j)≥nℓ+1(j)+2j and nmj(j)≥j+max(j−i+1,0)+2j(mj+1+⋯+mk−1), where mjmj is the number of parts in n(j)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−1k−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.