Combinatorial Families That Are Exponentially Far From Being Listable In Gray Code Sequence

Abstract

. Let S(n) be a collection of subsets of f1; :::; ng. In this paper we study numerical obstructions to the existence of orderings of S(n) for which the cardinalities of successive subsets satisfy congruence conditions. Gray code orders provide an example of such orderings. We say that an ordering of S(n) is a Gray code order if successive subsets differ by the adjunction or deletion of a single element of f1; : : : ; ng. The cardinalities of successive subsets in a Gray code order must alternate in parity. It follows that if d(S(n)) is the difference between the number of elements of S(n) having even (resp. odd) cardinality, then jd(S(n))j \Gamma 1 is a lower bound for the cardinality of the complement of any subset of S(n) which can be listed in Gray code order. For g 2, the collection B(n;g) of g-block free subsets of f1; : : : ; ng is defined to be the set of all subsets S of f1; : : : ; ng such that ja \Gamma bj g if a; b 2 S and a 6= b. We will construct a Gray code order for ..