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We study the total number of occurrences of several vincular (also calledgeneralized) patterns and other statistics, such as the major index and theDenert statistic, on permutations avoiding a pattern of length 3, extendingresults of Bona (2010, 2012) and Homberger (2012). In particular, for2-3-1-avoiding permutations, we find the total number of occurrences of anyvincular pattern of length 3. In some cases the answer is given by simpleexpressions involving binomial coefficients. The tools we use are bijectionswith Dyck paths, generating functions, and block decompositions ofpermutations.

Total occurrence statistics on restricted permutations