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Vincular and covincular patterns are generalizations of classical patternsallowing restrictions on the indices and values of the occurrences in apermutation. In this paper we study the integer sequences arising as theenumerations of permutations simultaneously avoiding a vincular and acovincular pattern, both of length 3, with at most one restriction. We seefamiliar sequences, such as the Catalan and Motzkin numbers, but also somepreviously unknown sequences which have close links to other combinatorialobjects such as lattice paths and integer partitions. Where possible we includea generating function for the enumeration. One of the cases considered settlesa conjecture by Pudwell (2010) on the Wilf-equivalence of barred patterns. Wealso give an alternative proof of the classic result that permutations avoiding123 are counted by the Catalan numbers.

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