The term combinatorial Gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some prespecified, small way. This notion generalizes the classical binary reflected Gray code scheme for listing n-bit binary numbers so that successive numbers differ in exactly one bit position, as well as work in the 1960s and 1970s on minimal change listings for other combinatorial families, including permutations and combinations.
The area of combinatorial Gray codes was popularized by Herbert Wilf in his invited address at the SIAM Conference on Discrete Mathematics in 1988 and his subsequent SIAM monograph [Combinatorial Algorithms: An Update, 1989] in which he posed some open problems and variations on the theme. This resulted in much recent activity in the area, and most of the problems posed by Wilf are now solved.
In this paper, we survey the area of combinatorial Gray codes, describe recent results, variations, and trends, and highlight some open problems.

A Survey of Combinatorial Gray Codes

Algorithms are presented to generate the n-bit binary reflected Gray code and codewords of fixed weight in that code. Both algorithms are efficient in that the time required to generate the next element from the current one is constant. Applications to the generation of the combinations of n things taken k at a time, the compositions of integers, and the permutations of a multiset are discussed.

Efficient generation of the binary reflected gray code and its applications

This paper surveys the numerous methods tha t have been proposed for pe rmuta tmn enumera t ion by computer. The various algori thms which have been developed over the years are described in detail, and zmplemented in a modern ALc, oL-hke language. All of the algori thms are derived from one rumple control structure. The problems involved with implement ing the best of the algori thms on real computers are t reated m detail. Assembly-language programs are derived and analyzed fully. The paper is intended not only as a survey of permuta t ion generation methods, but also as a tutomal on how to compare a number of different algori thms for the same task

Permutation Generation Methods

M. Douglas Mcllroy* and Data Structures Editor stop(top). In such a situation, the element top has to be deleted from the stack and more operations are required to generate the next combination. When k > top > 2, one can show that the probability for a specific value of top that a[top] = stop(top) is a(top + l ) /a( top) , which reduces to (k top + l ) / (n top). Hence, when k is small compared to n, it is very unlikely that the next combination is generated by using the theoretical maxim u m number of operations. I f k is very small compared to n, then P(I) , the probabili ty that a combination is generated by changing only a[1], is approximately 1. In this case, almost all the combinations are generated by the single statement a[1] := a[1] 1. It is doubtful that any combination algorithm would require less work than this to generate a combination.

The solution for the branching factor of the alpha-beta pruning algorithm and its optimality

A Chronicle of Permutation Statistical Methods

The purpose of this work is to find a method for building loopless algorithms for listing combinatorial items, like partitions, permutations, combinations. Gray code, etc. Algorithms for the above sequence are detailed in full.

Loopless Algorithms for Generating Permutations, Combinations, and Other Combinatorial Configurations

Algorithm 466: four combinatorial algorithm [G6]

1) are not changed until a(i) takes all available values: that is, if i > k then no other value but its present one, else, all values between I and rain(max,,, < ~ a(m) + 1, r). All those values are ordered in a sequence starting at 1 and ending at 2 (\"a 1-2 path\") or vice versa (\"a 2-1 path\"). Each sequence can be illustrated as moving a route of i along all available cells each time visiting one new cell. Each of the seven labels ONE... SEVEN, appearing in PARTEXACT, deals with a special segment of one of the two paths. It moves i to the appropriate new cell. ONE deals with the first move of an element initially located in the first cell. The roles of the other labels are illustrated in Figures 1 and 2. Each of the arrows describes the effect of the appropriate label. If i enters the cell z-+-I, we transfer k + 1 from that cell to the first one (from which it starts a 1-2 path). On the other hand, if the move of i empties its old cell, we transfer k to that cell. For each i, p(i) denotes the segment of the i's path according to which i Description Purpose. Procedure PARTEXACT produces, by successive calls, a sequence of all S(it,r) partitions of a set of n distinct elements into exactly r mutually exclusive subsets. (S(n,r) is the Stirling number of the second kind, see [1].) We assume that n >__ r > 2. There is no distinction of order: neither within subsets nor among them. We assume the elements to be numbers 1, 2,..., n. (If this is not the case, we just index the elements.) We also assume that we have a sequence of r numbered cells in which the subsets are located. The first cell contains the number 1 (together with the whole subset to which 1 belongs), then each cell contains the minimal element not contained in the preceding cells. Partitions are represented by an address-array, a, of n components. Every j is located in the cell numbered a(j). It follows that: 1. a(1) = 1, 2. a(j) < rain (max a(m) + 1, r). m<j After each call to PARTEXACT we receive a new address-array, a, which differs from the old one in, at most, two components. A new partition is received from the …

Gideon Ehrlich

Papers

Loopless Algorithms for Generating Permutations, Combinations, and Other Combinatorial Configurations

Algorithm 466: four combinatorial algorithm [G6]

Generator of set-partitions to exactly R subsets [G7]