Bijections between Partitions by Two-Directional Rewriting Techniques

Abstract

One basic activity in combinatorics is to establish combinatorial identities by so-called ‘bijective proofs,’ which consist in constructing explicit bijections between two types of the combinatorial objects under consideration. We show how such bijective proofs can be established, and how the 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. As a corollary, a new 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 novel two-directional reductions in the sense that forward and backward application of rewrite rules head 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.