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Universal cycles for permutations. (English) Zbl 1181.05005
A word $$u_1u_2\ldots u_{n!}$$ is called a universal cycle for $$S_n$$ if there is exactly one $$u_{i+1}u_{i+2}\ldots u_{i+n}$$ order-isomorphic to each permutation in $$S_n$$. The author shows how to construct a universal cycle for $$S_n$$ using only $$n+1$$ letters. This is best possible and proves a conjecture of F. Chung, P. Diaconis, and R. Graham [Discrete Math. 110, No.1-3, 43–59 (1992; Zbl 0776.05001)]. Moreover, bounds on the number of universal cycles for $$S_n$$ over the alphabet $$\{0,1,\ldots,n\}$$ are given.

##### MSC:
 05A05 Permutations, words, matrices
##### Keywords:
universal cycles; combinatorial generation; permutations
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##### References:
 [1] Chung, F.; Diaconis, P.; Graham, R., Universal cycles for combinatorial structures, Discrete math., 110, 43-59, (1993) · Zbl 0776.05001 [2] de Bruijn, N.G., A combinatorial problem, Nederl. akad. wetensch., proc., 49, 758-764, (1946) · Zbl 0060.02701 [3] Jackson, B.W., Universal cycles for $$k$$-subsets and $$k$$-permutations, Discrete math., 117, 141-150, (1993) · Zbl 0783.05001 [4] A.M. Williams, Shorthand universal cycles for permutations, in: ACM-SIAM Symposium on Discrete Algorithms 2008, 2007 (submitted for publication) · Zbl 1253.68273
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