Trotter, H. F. (1962), `Perm (algorithm 115)', Commun.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in part by the National Science Foundation Grant No. CCR8906500 1 elements. This was shown to be possible in several papers, including [26] 1] 2] and [9] which are described in [24] In fact, it is possible even if the two elements exchanged are required to be in adjacent positions ( 13] [25]) It is interesting to note that in the Wells Boothroyd Heap algorithms, the last permutation differs from the first permutation by a transposition only when n is odd, so the scheme is not cyclic . The Johnson Trotter scheme for generating permutations by adjacent transpositions is cyclic and is ....

....will be an edge in G[S] from p to q if and only if q can be obtained from p by a change specified in S. A question of interest, then, is whether or not G[S] has a Hamiltonian path or cycle. For example, by results cited above, G[S] is Hamiltonian when S allows only adjacent transpositions ( 13] [25]) or when S allows only derangements and n 6= 3 ( 7] 17] As another example, it is shown in [15] that for a set S of transpositions (of specific positions) G[S] is Hamiltonian if the elements of S form a basis for the symmetric group. A well known open problem in graph theory, due to Lov asz ....

H. F. Trotter, "PERM (Algorithm 115)", Comm. ACM, 5, No. 8 (1962), pp. 434-435.

....of the elements of S such that successive elements differ by a small amount. The classic example has the set S being the binary strings of length n and the small amount being the complementation of a single bit. Gray codes exist for a wide variety of combinatorial sets, including permutations [Tr62, Jo63], binary trees [LRR] integer partitions [Sa89, RSW] and linear extensions of some posets [Ru92, PR91] In this paper, we study Gray codes for acyclic orientations of graphs. Let G = V (G) E(G) be an undirected graph. If we replace each edge (u; v) 2 E(G) with either the arc u v or the arc ....

....basic combinatorial sets are binary strings and permutations. There are well known Gray codes for listing both of these sets. The original binary reflected Gray code [Gr53] lists all binary strings of length m such that adjacent strings differ by a single bit. There is a Gray code for permutations [Tr62, Jo63] which lists all permutations of m items such that adjacent permutations differ by a single transposition. Both of these Gray codes are cyclic in that the first and last elements listed also differ in the prescribed way. In this section, we show that both of these Gray codes are generalized by the ....

H. F. Trotter, "PERM (Algorithm 115)," Communications of the ACM, 8 (1962) 434-435. 21

....the removal of ffi ones) where, in the case of D(n; k) ffi = 1. 3 1 Introduction Recent work in combinatorial enumeration has considered listing special sets so successive elements differ by a small, pre specified change. Examples include (1) generating permutations by adjacent transpositions [5, 16] (2) generating bit strings by changing one bit [4, 3] 3) generating subsets by changing one element [1, 8, 12] 4) generating binary trees by rotations [7] 5) generating Coxeter group elements by reflection [2] and (6) generating linear extensions of certain posets by transpositions [9, 10, ....

H. F. Trotter, "PERM (Algorithm 115)", Communications of the ACM 5, No. 8 (1962) 434-435.

....one bit. By applying the binary Gray code, a variety of problems have been solved and the complexities of the solutions to other problems have been improved [Gar, ChLeDu, ChChCh, Los, Ric] There are many examples of combinatorial families for which Gray codes are known, including permutations [Joh, Tro], combinations [BuWi, NiWi, Rus1] compositions [Kli] set partitions [Kay] integer partitions [Sav, RaSaWe] binary trees [RuPr, Luc, LuRoRu] and linear extensions [PrRu1, PrRu2, Rus2, Sta, Wes] When an application requires an exhaustive examination of all objects in a combinatorial family, ....

H. F. Trotter, "PERM (Algorithm 115)", Communications of the ACM 5, 8 (1962) 434-435.

....of Lov asz that every connected, undirected, vertex transitive graph has a Hamilton path [Lo] It is even unknown whether every connected Cayley graph of S n is hamiltonian. Some results on Cayley graphs are surveyed in [WiGa] It was established independently by Johnson [Jo] and Trotter [Tr] that it is possible to generate all permutations of 1 : n, each exactly once, so that successive permutations (as well as the first and last) differ only by one swap of two elements in adjacent positions, that is, by an adjacent transposition. This gives an efficient algorithm for generating ....

H. F. Trotter, "PERM (Algorithm 115)", Communications of the ACM 5 (1962) 434-435.

.... at a permutation with a j in position k for any j; k 2 [n] Tchuente generalized both of these results by showing that any two permutations of different parity are joined by a Hamilton path in Cay(X : S n ) Tc] As an example, the well known algorithm of Steinhaus [St] Johnson [Jo] and Trotter [Tr], for generating permutations by adjacent transpositions, gives a Hamilton cycle through the Cayley graph of S n with generating set f(12) 23) 34) n Gamma 1 n)g. However, an element of S n of order two need not be a transposition, so it remains open whether the Cayley graph of S n ....

H. F. Trotter, "PERM (Algorithm 115)", Communications of the ACM 5, No. 8 (1962) 434-435.

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Trotter, H. F. (1962), `Perm (algorithm 115)', Commun.

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L. Trotter, "PERM (Algorithm 115)", Communications of the ACM 5 (1962).

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L. Trotter, "PERM (Algorithm 115)", Communications of the ACM 5 (1962).

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