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

Abstract

A bijection is presented between (1): partitions with conditions f j + f j + 1 ≤ k − 1 and f 1 ≤ 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 n ℓ ( j ) ≥ n ℓ + 1 ( j ) + 2 j and n m j ( j ) ≥ j + max ( j − i + 1 , 0 ) + 2 j ( m j + 1 + ⋯ + m k − 1 ) , 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.