Pattern expansions of permutation statistics

We study the expansions of permutation statistics in the basis of functions counting occurrences of a fixed pattern in a permutation. We show the finiteness of these pattern expansions for a class of permutation statistics including the higher moment statistics, generalizing a result of Berman and Tenner. We also give a combinatorial criterion for the positivity of pattern expansions. Using this criterion, we show that the pattern expansion of the number of reduced words of a permutation is positive and give an enumerative interpretation for the coefficients.