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

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 [3], later, Ruskey presented a modified version of Knuth’s algorithm with distance two, Ehrlich [5] introduced a loop-free algorithm for the set of partitions of {1, 2, . . . , n}, Ruskey and Savage [16] generalized Ehrlich’s results and give two Gray codes for the set of partitions of {1, 2, . . . , n}, and recently, Mansour et al. [11] 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.