Refining Enumeration Schemes to Count According to Permutation Statistics

Abstract

We develop algorithmic tools to compute quickly the distribution of permutation statistics over sets of pattern-avoiding permutations. More specfically, the algorithms are based on enumeration schemes, the permutation statistics are based on the number of occurrences of certain vincular patterns, and the permutations avoid sets of vincular patterns. We prove that whenever a finite enumeration scheme exists to count the number of pattern-avoiding permutations, then the distribution of statistics like the number of descents can also be computed based on the same scheme. Statistics such as the number of peaks, right-to-left maxima, and the major index are also investigated, as well as multi-statistics.