Math 630 — Fall 2010
Homework 2
Due Monday, October 11, in class.
Problem 1. A partition is self-conjugate if equals its conjugate partition. Show that the number of
self-conjugate partitions of n equals the number of partitions of n with all parts odd and distinct.
Problem 2.
1. Show that the number of paths from (0, 0) to (m, n) using steps (1, 0) and (0, 1) is m+n
.
m
P |λ|
2. Let Amn (q) =
q , where the sum is over partitions with at most m parts, whose largest
.
part is at most n. Show that Amn (1) = m+n
m
3. Show that Amn (q) = m+n
. Hint: Show that Amn (q) satisfies the same recurrence and
m q
initial conditions as the q-binomial coefficient.
Problem 3. Prove that
n X
i
i=0
Problem 4. The sum Gn =
Pn
k
n
k q
k=0
q
q
(k+1)(n−i)
n+1
=
.
k+1 q
is called a Galois number. Show that
Gn+1 = 2Gn + (q n − 1)Gn−1 .
Problem 5. Recall that S(n, k) is the Stirling number of the second kind. Prove
n X
n
S(n + 1, k + 1) =
S(i, k).
i
i=0
P a1 −1 a2 −1
Problem 6. Show that S(n, k) =
1
2
· · · k ak −1 , where the sum is over all solutions of
a1 + a2 + · · · + ak = n in positive integers.
Problem 7. Show that S(n, k) is the number of sequences a1 a2 . . . an of positive integers such
that the largest entry is k, and the first occurrence of i appears before the first occurrence of i + 1
(1 ≤ i ≤ k − 1).
Problem 8. Give a simple formula for the number of partitions of n with parts ≥ 2 using partition
numbers.
Problem 9. A partition of n is called perfect if it contains precisely one partition for every m < n.
For example, 311, 221, and 11111 are the perfect partitions of 5. Show that the number of perfect
partitions of n equals the number of ordered factorizations of n + 1 without unit factors.
Problem 10. Let e(n), o(n) and sc(n) denote, respectively, the number of partitions of n with an
even number of even parts, with an odd number of even parts, and that are self conjugate. Show
that e(n) − o(n) = sc(n).