Enumeration of up-down permutations by number of rises
L. Carlitz · Pacific Journal of Mathematics · 1973
It is well known that A(ri), the number of up-down permutations of {1, 2, , n} satisfies In the present paper generating functions are obtained for the number of up-down permutations counting the number of rises among the "peaks".1* If (a lf θi, •••,«») denotes an arbitrary up-down permutation, then (b u δ 2 , •••, b n ), where &. = n -at + 1is a down-up permutation and vice versa.