Two Reflected Gray Code-Based Orders on Some Restricted Growth Sequences

Abstract

We consider two order relations: that induced by the m-ary reflected Gray code and a suffix partitioned variation of it. We show that both of them when applied to some sets of restricted growth sequences still yield Gray codes. These sets of sequences are: subexcedant or ascent sequences, restricted growth functions, and staircase words. In each case we give efficient exhaustive generating algorithms and compare the obtained results.Remark 1. If st is a statistic satisfying relations (2) and (3) above, then 1. max{st(s 1 s 2 . . . s n ) | s 1 s 2 . . . s n is an st-restricted growth sequence} = n − 1;

More Restricted Growth Functions: Gray Codes and Exhaustive Generation

“…This order relation is the natural extension to m-ary sequences of the order induced by the Binary Reflected Gray Code introduced in [2]. See for example [1,4] where this order relation and its variations are considered in the context of factor avoiding words and of statistic-restricted sequences.…”

“…In the present paper we investigate two prefix partitioning order relations on the set of bounded restricted growth functions: as in [4], the original Reflected Gray Code Order on m-ary sequences, and a new order relation which is an appropriate modification of the former one. We show that, according to the parity of the imposed bound, one of these order relations gives a Gray code on the set of bounded restricted growth functions.…”

“…In [4] the authors shown that both the order relation induced by the generalization of the Binary Reflected Gray Code and one of its suffix partitioned version yield Gray codes on some sets of restricted integer sequences, and in particular for restricted growth functions. These results are presented in a general framework, where the restrictions are defined by means of statistics on integer sequences.…”

On the exhaustive generation of generalized ballot sequences in lexicographic and Gray code order

“…Here we show that the restriction of the Reflected Gray Code to the sets of length n generalized and p-ary, with p even, ballot sequences induces a 3-adjacent Gray code. Similar techniques based on variations of the order relation induced by the Reflected Gray Code was used implicitly, for example in [5,19], and developed systematically as a general method in [1,11,12,[16][17][18], and our Gray codes are in the light of this direction. In the final part of this paper we give constant amortized time exhaustive generating algorithms for these classes of ballot sequences, in lexicographic order and for the corresponding Gray codes.…”

Gray code orders for $$q$$ q -ary words avoiding a given factor

“…Here, we adopt a different approach by relaxing Squire's 'one position constraint' and give Gray codes for length n words avoiding any given factor, where consecutive words differ in at most three positions. Our definitions for these Gray codes are based on two order relations inspired from the original Binary Reflected Gray Code [5]; similar techniques were used previously (less or more explicitly) for other combinatorial classes, see for example [18,11,15] and the references therein. More precisely, we characterize forbidden factors inducing zero periodicity (defined later), which is a crucial notion for our construction; and we show that the zero periodicity property of a forbidden factor is a sufficient condition for the set of words avoiding this factor when listed in the appropriate order to be a Gray code.…”