MATH 630. Combinatorics, Fall 2008
Assignment 2. Due on Wednesday, September 24, 2008
1. Let A(n, k) be the Eularian number, that is, A(n, k) is the number of
permutations of length nwith k−1 descents. Prove that A(n, k) satisfies
the recurrence
A(n, k) = kA(n−1, k )+(n−k+ 1)A(n−1, k −1),for 2 ≤k≤n,
with boundary conditions A(n, 0) = 0, A(n, 1) = A(n, n) = 1 and A(n, k) =
0 for k > n.
2. Again A(n, k) is the Eularian number. Prove that
A(n, k) = A(n, n + 1 −k).
3. Let S⊆[n−1], and let α(S) denote the number of n-permutations
whose descent set is contained in S.
Find the one-element set {i} ⊆ [n−1] for which α({i}) is maximal.
4. Prove that for any permutation π,i(π) = i(π−1), where i(π) is the
number of inversions of π.
5. A permutation πis called even if i(π) is even. Similarly, πis called odd
if i(π) is odd. Let n≥2. Prove that the number of even (odd) permutations
of length nis n!/2.
6.Exercises on page 49, Problem 31. For a permutation π, let m(π)
denote the number of left-to-right maxima of πand i(π) the number of
inversions of π. Compute the generating function
F(x, q) = X
π∈Sn
xm(π)qi(π).
Please state your reason.
7. The order of a permutation πis the smallest positive integer kfor
which πk=id. Assume that pi is of cycle type (c1, c2, . . . , cn). What is the
order of π?
8. How many permutations has length 6 whose fourth power is the
identity permutation?