Gray Codes for A-Free Strings

Abstract

For any $q \geq 2$, let $\Sigma_{q}=\{0,\ldots,q\!-\!1\}$, and fix a string $A$ over $\Sigma_{q}$. The $A$-free strings of length $n$ are the strings in $\Sigma_{q}^n$ which do not contain $A$ as a contiguous substring. In this paper, we investigate the possibility of listing the $A$-free strings of length $n$ so that successive strings differ in only one position, and by $\pm 1$ in that position. Such a listing is a Gray code for the $A$-free strings of length $n$. We identify those $q$ and $A$ such that, for infinitely many $n \geq 0$, a Gray code for the $A$-free strings of length $n$ is prohibited by a parity problem. Our parity argument uses techniques similar to those of Guibas and Odlyzko (Journal of Combinatorial Theory A 30 (1981) pp. 183–208) who enumerated the $A$-free strings of length $n$. When $q$ is even, we also give the complementary positive result: for those $A$ for which an infinite number of parity problems do not exist, we construct a Gray code for the $A$-free strings of length $n$ for all $n \geq 0$.

A Survey of Combinatorial Gray Codes

“…, N i − 1. Squire generalizes results on clean words to m-ary Gray codes [Squ96], but leaves open the case when m is odd.…”

“…, n}, ordered by inclusion, and let H n denote the Hasse diagram of B n . The correspondence shown that the answer is yes if α can be written as α = ββ · · · β, where β is a string with the property that no nontrivial prefix of β is also a suffix of β; otherwise there are parity problems for infinitely many n [Squ96].…”

A trace partitioned Gray code forq-ary generalized Fibonacci strings

“…A consequence of the expansion technique is that our Gray code has the following property: if we replace each non-zero symbol in each string by 1, and 'collapse' the obtained list by keeping one copy of each binary strings, then the existing Gray code for q = 2 is obtained. The investigation on the existence of Gray codes for strings on a q-ary alphabet avoiding a general consecutive pattern has already been studied: in [8] the author gives such a Gray code only when q even, while the case of q odd is left open. Our Gray code deals with the avoidance of a particular pattern but works for any q, and an interesting development could be a deeper investigation to check if this constructions can be applied to a general consecutive pattern in order to solve the open question in [8].…”

“…The investigation on the existence of Gray codes for strings on a q-ary alphabet avoiding a general consecutive pattern has already been studied: in [8] the author gives such a Gray code only when q even, while the case of q odd is left open. Our Gray code deals with the avoidance of a particular pattern but works for any q, and an interesting development could be a deeper investigation to check if this constructions can be applied to a general consecutive pattern in order to solve the open question in [8]. Also, it would be of interest to implement our Gray code definition into an efficient generating algorithm for the set of underlying strings.…”

Gray codes for Fibonacci q-decreasing words

“…Over the time, the study of words and patterns became more abstract and systematic (see for instance Lothaire's books [12,13,14] and [3]). An important amount of questions concerning efficient enumeration and generation of words respecting certain properties (including pattern avoidance) were mathematically formulated and answered only relatively recently, the works closely related to the present study include [1,2,4,5,7,20,21,22].…”