Solution of some multi-dimensional lattice path parity difference recurrence relations

Abstract

Many elementary combinatorial objects can be modeled as restricted multiset permutations. Standard examples include (unrestricted) permutations of a multiset and ordered trees with prescribed degree sequence. In developing algorithms to generate all members of a subset S of multiset permutations in such a way that successive permutations differ only by the interchange of adjacent distinct elements, we are led to the adjacent interchange graph G( S) of S. The vertices of G( S) are the elements of S and two vertices are connected by an edge if one can be obtained from the other by an adjacent interchange. This graph is bipartite. no adjacent interchange algorithm is possible if the number of vertices in the two bipartitions differs by more than one. In this paper we determine a recurrence relation that describes the difference of the number of vertices in the two bipartitions. This recurrence relation is solved for the examples mentioned above.