Value-Peaks of Permutations

In this paper, we focus on a “local property” of permutations: value-peak. A permutation σ has a value-peak σ(i) if σ(i − 1) < σ(i) > σ(i + 1) for some i ∈ [2, n − 1]. Define V P (σ) as the set of value-peaks of the permutation σ. For any S ⊆ [3, n], define V Pn(S) such that V P (σ) = S. Let Pn = {S | V Pn(S) 6= ∅}. we make the set Pn into a poset Pn by defining S T if S ⊆ T as sets. We prove that the poset Pn is a simplicial complex on the set [3, n] and study some of its properties. We give enumerative formulae of permutations in the set V Pn(S). Partially supported by NSC 98-2115-M-110-009 Corresponding author Partially supported by NSC 96-2115-M-001-005 the electronic journal of combinatorics 17 (2010), #R46 1