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On multi-avoidance of generalized patterns

Kitaev, Sergey and Mansour, Toufik (2005) On multi-avoidance of generalized patterns. Ars Combinatoria, 76. pp. 321-350.

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In [Kit1] Kitaev discussed simultaneous avoidance of two 3-patterns with no internal dashes, that is, where the patterns correspond to contiguous subwords in a permutation. In three essentially different cases, the numbers of such n-permutations are 2n−1, the number of involutions in n, and 2En, where En is the n-th Euler number. In this paper we give recurrence relations for the remaining three essentially different cases. To complete the descriptions in [Kit3] and [KitMans], we consider avoidance of a pattern of the form x−y−z (a classical 3-pattern) and beginning or ending with an increasing or decreasing pattern. Moreover, we generalize this problem: we demand that a permutation must avoid a 3-pattern, begin with a certain pattern and end with a certain pattern simultaneously. We find the number of such permutations in case of avoiding an arbitrary generalized 3-pattern and beginning and ending with increasing or decreasing patterns.