Journal of Integer Sequences, Vol. 20 (2017), Article 17.7.6

Enumerations of Permutations Simultaneously Avoiding a Vincular and a Covincular Pattern of Length 3

Christian Bean and Henning Ulfarsson
School of Computer Science
Reykjavik University
Menntavegi 1
101 Reykjavik

Anders Claesson
Science Institute
University of Iceland
Dunhaga 5
107 Reykjavik


Vincular and covincular patterns are generalizations of classical patterns allowing restrictions on the indices and values of the occurrences in a permutation. In this paper we study the integer sequences arising as the enumerations of permutations simultaneously avoiding a vincular and a covincular pattern, both of length 3, with at most one restriction. We see familiar sequences, such as the Catalan and Motzkin numbers, but also some previously unknown sequences which have close links to other combinatorial objects such as lattice paths and integer partitions. Where possible we include a generating function for the enumeration. One of the cases considered settles a conjecture by Pudwell (2010) on the Wilf-equivalence of barred patterns. We also give an alternative proof of the classic result that permutations avoiding 123 are counted by the Catalan numbers.

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(Concerned with sequences A000079 A000108 A000124 A001006 A098569 A121690 A249560 A249561 A249562 A249563.)

Received February 14 2017; revised versions received June 9 2017; July 3 2017. Published in Journal of Integer Sequences, July 5 2017.

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