Constructive techniques for labeling constant weight Gray codes with applications to minimal generating sets of semigroups

Abstract

The Partition Type Conjecture, a generalization of the Middle Levels Conjecture of combinatorics, states that for all positive integers n and r, with n> r>1, and every non-exceptional partition type τ of weight r of X n , there exists a constant weight Gray code which admits an orthogonal labeling by partitions of type τ. We prove results in combinatorics and finite semigroup theory, providing the completion of the proof that the Partition Type Conjecture is true for all types having more than one class of size greater than one (and leaving open only those cases which are equivalent to the Middle Levels Conjecture). The rank of a finite semigroup S is the cardinality of a minimum generating set for S; if S is idempotent generated, the idempotent rank of S is the cardinality of a minimum idempotent generating set for S. A semigroup of transformations of X n ={1,…, n} is said to be S n -normal if S is closed under conjugation by the permutations of X n . The results here concerning the Partition Type Conjecture are used to determine a simple formula for the rank and idempotent rank of every S n -normal semigroup.