Meyerowitz, Aaron

The topic of this paper is tiling the integers with triples, or more precisely to write
Z as a disjoint union of translates of a given set of 3-subsets composed of basic shapes
called prototiles. We fix the set of proto tiles P = { { 0, a, a+ v} , { U. b, a+ b}} and define
an algorithm which returns a sequence of translates of P when given an initial subset
of Z representing integers that are already tiled. This algorithm is then adapted to
describe all possible tilings with triples from P using the action of certain signed
permutation matrices on a subset of za+b , uamdy the 2" Yectors with all entries ±1.
Given b > 2a, we research properties of the digraph of all possible tiling states and
some related digraphs.

This paper surveys work of the last few years on construction of bijections for
partition identities. We use the more general setting of sieve--equivalent families. Suppose A1' ... ,An are subsets of a finite set A and B1' ... ,Bn are subsets of a finite
set B. Define AS=∩(i∈S) Ai and BS = ∩ (i∈S) Bi for all S⊆N={1,...,n}. Given explicit bijections fS: AS->BS for each S⊆N, A-∪Ai has the same size as B-∪Bi. Several
authors have given algorithms for producing an explicit bijection between these two
sets. In certain important cases they give the same result. We discuss and
compare algorithms, use Graph Theory to illustrate them, and provide PAS CAL
programs for them.