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

Abstract

A bijection is presented between (1): partitions with conditions $f_j+f_{j+1}\le k-1$ and $f_1\le i-1$, 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 $n_{\ell }^{(j)}\ge n_{\ell +1}^{(j)}+2j$ and $n_{m_j}^{(j)}\ge j+\max (j-i+1,0)+2j(m_{j+1}+...+m_{k-1})$, 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.

12 pages; minor corrections, version to appear in Discrete Math

Cited by in corpus (1)

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