TAILIEUCHUNG - Báo cáo khoa học: An answer to a question by Wilf on packing distinct patterns in a permutation

There has been significant interest in the topic of finding permutations containing many copies of the same pattern. In this paper, we will be concerned with the other extremity, permutations containing as many different patterns as possible. At the Conference on Permutation Patterns, Otago, New Zealand, 2003, Herb Wilf asked how many distinct patterns could be contained in a permutation of length n. Based on empirical evidence, it seemed this number may approach the theoretical upper bound of 2n. In this paper we enumerate patterns contained in each of a certain class of permutations to at least establish a lower bound for this function | An answer to a question by Wilf on packing distinct patterns in a permutation Micah Coleman Submitted Apr 21 2004 Accepted May 12 2004 Published May 24 2004 MR Subject Classifications 05A05 05A16 Abstract We present a class of permutations for which the number of distinctly ordered subsequences of each permutation approaches an almost optimal value as the length of the permutation grows to infinity. 1 Introduction Definition Let q q1q2 . qk E Sk be a permutation and let k n. We say that the permutation p p1p2 pn E Sn contains the pattern q if there is a set of indices 1 ii i2 ik n such that pi1 pi2 pik. There has been significant interest in the topic of finding permutations containing many copies of the same pattern. In this paper we will be concerned with the other extremity permutations containing as many different patterns as possible. At the Conference on Permutation Patterns Otago New Zealand 2003 Herb Wilf asked how many distinct patterns could be contained in a permutation of length n. Based on empirical evidence it seemed this number may approach the theoretical upper bound of 2n. In this paper we enumerate patterns contained in each of a certain class of permutations to at least establish a lower bound for this function. Let f p be the number of distinct patterns contained in a permutation p. Let h n be the maximum f p where the maximum is taken over all permutations of length n. Department of Mathematics University of Florida Gainesville FL 32611-8105. Supported by a grant from the University of Florida University Scholars Program mentored by Miklos Bona. THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 N8 1 On the one hand Pn 0 k 2n is an obvious upper bound for h n . On the other hand there are only k patterns of length k. So for small k we can replace n with k . As n grows this second bound quickly becomes insignificant as k for all k above a breakpoint which grows much slower than n. Wilf demonstrated a class of permutations Wn for which f Wn .

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