Combinatorial families that are exponentially far from being listable in Gray code sequence

Abstract

Abstract. Let S(n) be a collection of subsets of {1, ..., n}. 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 {1, . . . , n}. 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 |d(S(n))| − 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-blockfree subsets of {1, . . . , n} is defined to be the set of all subsets S of {1, . . . , n} such that |a − b| ≥ g if a, b ∈ S and a = b. We will construct a Gray code order for B(n, 2). In contrast, for g > 2 we find the precise (positive) exponential growth rate of d(B(n, g)) with n as n → ∞. This implies B(n, g) is far from being listable in Gray code order if n is large. Analogous results for other kinds of orderings of subsets of B(n, g) are proved using generalizations of d (B(n, g)) . However, we will show that for all g, one can order B(n, g) so that successive elements differ by the adjunction and/or deletion of an integer from {1, . . . , n}.We show that, over an A-letter alphabet, the words of length n which contain no block of k consecutive letters cannot, in general, be listed so that successive words differ by a single letter. However, if k > 2 and A > 2 or if k = 2 and A > 3, such a listing is always possible.

Combinatorial Gray Codes — an Updated Survey

“…Subsequently, Vajnovszki [Vaj01] presented a simple genlex Gray code for Fibonacci words; see Figure 3 (b) and also [MS05]. Chinburg, Savage and Wilf [CSW99] consider bitstrings in which any two 1s are at least d ⩾ 2 positions apart. For d = 2 these are precisely Fibonacci words (with ℓ = 2), and the paper describes the same genlex Gray code as Vajnovszki for this case.…”

“…Similarly, words avoiding any number of distinct factors all starting with 1 are a flip language. As an application, recall that Chinburg, Savage and Wilf [CSW99] considered bitstrings in which any two 1s are at least d ⩾ 2 positions apart. Those are characterized by avoiding the factors {11, 101, 1001, .…”

Combinatorial Gray codes-an updated survey

“…Subsequently, Vajnovszki [Vaj01] presented a simple genlex Gray code for Fibonacci words; see Figure 3 (b) and also [MS05]. Chinburg, Savage and Wilf [CSW99] consider bitstrings in which any two 1s are at least d ≥ 2 positions apart. For d = 2 these are precisely Fibonacci words (with = 2), and the paper describes the same genlex Gray code as Vajnovszki for this case.…”