Math 630 — Fall 2010
Homework 1
Due Monday, September 27, in class.
Note: You may use any kind of proof except when explicitly instructed to use a combinatorial
proof.
Problem 1.
1. Give a combinatorial proof for
Pn
k=0
2k = 2n+1 − 1.
Pn
− k)2k−1 .
m
n−k n
n
3. Give a combinatorial proof for m
=
, where n ≥ m ≥ k ≥ 0, and use this to
k
k m−k
Pm n n−k compute k=0 k m−k .
2. Find (and prove) a formula for
k=1 (n
Problem 2. Let f (n, k) be the number of k-subsets of [n] that do not contain a pair of consecutive
integers. Show that f (n, k) = n−k+1
.
k
n
n
k+1
; k = n−k
and use this to verify that the sequence
Problem 3. Show that nk = nk n−1
k−1
k+1
n
, 0 ≤ k ≤ n, is unimodal, that is:
k
n
n
n
n
n
<
< ··· <
=
> ··· >
.
0
1
bn/2c
dn/2e
n
Problem 4. Prove that
n
n1 ,n2 ,n3
≤
3n
n+1
for n ≥ 1. Hint: Use Induction.
Problem 5. Recall that c(n, k) is the signless Stirling number of the first
Pnkind,i that counts permutations of an n-set with exactly k cycles. Show that c(n + 1, k + 1) = i=0 k c(n, i).
Problem 6. A permutation π ∈ Sn is an involution if π 2 is the identity permutation, that is, if all
cycles have length one or two. Set i0 = 1 and let in be the number of involutions of an n-set. Show
that in+1 = in + nin−1 .
Problem 7. Let ` > n2 . Show that the number of permutations π ∈ Sn that have a cycle of length
` equals n!` .
Problem 8. Let In,k be the number of permutations π ∈ Sn with exactly k inversions. Prove:
1. In,0 = 1.
2. In,k = In,(n)−k .
2
3. In,k = In−1,k + In,k−1 for k < n. Is this also true for k = n?
4.
P(n2 )
k
k=0 (−1) In,k
= 0 for n ≥ 2.
The In,k are called inversion numbers
Problem 9. Recall that A(n, k) is the Eulerian number that counts the permutations of an n-set
that have k − 1 descents.
1. Prove that A(n, k) = (n − k + 1)A(n − 1, k − 1) + kA(n − 1, k) for n, k ≥ 1, with initial
conditions A(0, 0) = 1 and A(0, k) = 0 for k > 0.
P
2. Use Induction to prove that xn = nk=0 A(n, k) x+n−k
.
n
Problem 10. Use the previous exercise to show that A(n, k) =
Pk
i n+1
i=0 (−1)
i
(k − i)n .