Explicit expressions for the extremal excedance set statistics

The excedance set of a permutation π=π1π2⋯πkπ=π1π2⋯πk is the set of indices ii for which πi>iπi>i. We give explicit formulas for the number of permutations whose excedance set is the initial segment {1,2,…,m}{1,2,…,m} and also of the form {1,2,…,m,m+2}{1,2,…,m,m+2}. We provide two proofs. The first is an explicit combinatorial argument using rook placements. The second uses the chromatic polynomial and two variable exponential generating functions. We then recast these explicit formulas as LDULDU-decompositions of associated matrices and show that these matrices are totally non-negative.