Finding direct partition bijections by two-directional rewriting techniques

Abstract

One basic activity in combinatorics is to establish combinatorial identities by so-called ‘bijective proofs,’ which consists in constructing explicit bijections between two types of the combinatorial objects under consideration. We show how such bijective proofs can be established in a systematic way from the ‘lattice properties’ of partition ideals, and how the desired bijections are computed by means of multiset rewriting, for a variety of combinatorial problems involving partitions. In particular, we fully characterizes all equinumerous partition ideals with ‘disjointly supported’ complements. This geometrical characterization is proved to automatically provide the desired bijection between partition ideals but in terms of the minimal elements of the order filters, their complements. As a corollary, a new transparent proof, the ‘bijective’ one, is given for all equinumerous classes of the partition ideals of order 1 from the classical book “The Theory of Partitions” by G.Andrews. Establishing the required bijections involves two-directional reductions technique novel in the sense that forward and backward application of rewrite rules heads, respectively, for two different normal forms (representing the two combinatorial types). It is well-known that non-overlapping multiset rules are confluent. As for termination, it generally fails even for multiset rewriting systems that satisfy certain natural invariant balance conditions. The main technical development of the paper (which is important for establishing that the mapping yielding the combinatorial bijection is functional) is that the restricted two-directional strong normalization holds for the multiset rewriting systems in question.