Combinatorial Gray Codes

Abstract

We consider families $\{ {\bf C}(n,k):O \leqq k \leqq n\} $ where each ${\bf C}(n,k)$ is a set of combinatorial objects, $C(n,k) = |{\bf C}(n,k)|$ satisfies a recursion $C(n,k)= a_{n,k}C(n - 1,k - 1) + b_{n,k} C(n - 1,k)$, and each object in ${\bf C}(n,k)$ is represented by an n-vector. We study “loop-free” or “uniformly bounded transition” algorithms, i.e., algorithms which yield linear orders on the sets ${\bf C}(n,k)$ so that the vectors representing consecutive objects are “close to each other” (combinatorial Gray codes).Key words. listing algorithms, uniformly bounded operations, uniformly bounded transition algorithms, loop-free algorithms, binary reflected Gray codes, combinatorial Gray codes, binomial grids