Enumeration of rises and falls by position

AbstractLet π=(π1, π2,…,πn) denote a permutation of Zn = {1, 2,…, n}. The pair (πi, πi+1) is a rise if πi<πi+1 or a fall if πi>πi+1. Also a conventional rise is counted at the beginning of π and a conventional fall at the end. Let k be a fixed integer ≥ 1. The rise πi,πi+1 is said to be in a in a j (mod k) position if i ≡ j (mod k); similarly for a fall. The conventional rise at the beginning is in a 0 (mod k) position, while the conventional fall at the end is in an n (mod k) position. Let Pn≡Pn(r0,…,rk−1,ƒ0,…,ƒ;k−1) denote the number of permutations having ri rises i (mod k) positions and ƒ;i falls in i (mod k) positions. A generating function for Pn is obtained. In particular, for k = 2 the generating function is quite explicit and also, for certain special cases when k = 4