Some results on the avoidance of vincular patterns by multisets

Vincular or dashed patterns resemble classical patterns except that some of the letters within an occurrence must satisfy an adjacency requirement. In this paper, we show some general equivalences concerning the avoidance of vincular patterns by multiset permutations. We prove our results by defining bijections between various avoidance classes that preserve the number of occurrences of each letter. As a consequence, we obtain for multiset permutations the complete Wilf-classification of patterns of type ( 2 , 1 , 1 ) , which also yields the complete classification for compositions and k -ary words when taken with numerical evidence.