The sorting index and equidistribution of set-valued statistics over restricted permutations

Abstract

We prove that the pairs of permutation statistics (sor,Cyc) and (inv,Rmil) are equidistributed on the set of permutations that correspond to arrangements of n non-atacking rooks on a fixed Ferrers board with n rows and n columns and give their generating functions. Our results extend recent results of Petersen. The key elements in the proofs are the A-code and the B-code introduced by Foata and Han. We show that the map B-code−1∘A-code is a bijection on the set of restricted permutations which sends inv to sor, Rmil to Cyc and preserves the set-valued statistics Lmal and Lmap. We also show analogous equidistribution results for restricted permutations of type B and D by constructing appropriate A-codes and B-codes in each case.