Loop-free Gray code algorithm for the e-restricted growth functions

Abstract

The subject of Gray codes algorithms for the set partitions of { 1 , 2 , … , n } had been covered in several works. The first Gray code for that set was introduced by Knuth (1975) [5], later, Ruskey presented a modified version of Knuthʼs algorithm with distance two, Ehrlich (1973) [3] introduced a loop-free algorithm for the set of partitions of { 1 , 2 , … , n } , Ruskey and Savage (1994) [9] generalized Ehrlichʼs results and give two Gray codes for the set of partitions of { 1 , 2 , … , n } , and recently, Mansour et al. (2008) [7] gave another Gray code and loop-free generating algorithm for that set by adopting plane tree techniques. In this paper, we introduce the set of e-restricted growth functions (a generalization of restricted growth functions) and extend the aforementioned results by giving a Gray code with distance one for this set; and as a particular case we obtain a new Gray code for set partitions in restricted growth function representation. Our Gray code satisfies some prefix properties and can be implemented by a loop-free generating algorithm using classical techniques; such algorithms can be used as a practical solution of some difficult problems. Finally, we give some enumerative results concerning the restricted growth functions of order d.