A pattern is said to be covincular if its inverse is vincular. In this paper we count the number of permutations simultaneously avoiding a vincular and a covincular pattern, both of length 3. 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 ascent sequences, lattice paths and integer partitions. Where possible we include a generating function for the enumeration. We also give an alternative proof of the classic result that permutations avoiding 123 are counted by the Catalan numbers.

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