10 research outputs found
An explicit formula for the number of permutations with a given number of alternating runs
Let denote the number of permutations of with
alternating runs. In this note we present an explicit formula for the numbers
.Comment: 6 page
Counting permutations by cyclic peaks and valleys
In this paper, we study the generating functions for the number of permutations
having a prescribed number of cyclic peaks or valleys. We derive
closed form expressions for these functions by use of various algebraic methods.
When restricted to the set of derangements (i.e., fixed point free permutations),
the evaluation at −1 of the generating function for the number
of cyclic valleys gives the Pell number. We provide a bijective proof of this
result, which can be extended to the entire symmetric group.
Keywords: Derangements; Involutions; Pell numbers; Cyclic valley