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Math 146: Algebraic Combinatorics
Homework 7
This problem set is due Friday, May 14.
Do problems 3.11 and 3.12, in addition to the following:
GK7.1.
The aim of this exercise is yet another proof of the exponential formula. Let
A
(
x
) =
∞
∑
k
=
1
a
k
x
k
k
!
be the exponential generating function for some types of animals of positive sizes. You’re
suppose to prove everything in this exercise directly, rather than by using theorems in chapter
3.
(a)
Show that there are
∑
j
+
k
+
`
=
n
±
j
+
k
+
`
j
,
k
,`
²
a
j
a
k
a
l
menageries of size
n
with three
numbered
animals.
(b)
Show that
A
(
x
)
3
is the e.g.f. for menageries with three numbered animals, more gener
ally that
A
(
x
)
n
is the e.g.f. for menageries with
n
numbered animals, and ﬁnally that
A
(
x
)
n
/
n
! is the e.g.f. for menageries with
n
unnumbered animals.
(c)
Use part (b) to prove that
M
(
x
) =
exp
A
(
x
)
.
GK7.2.
Let
f
n
be the number of partitions of
[
n
]
such that the number of parts is divisible by 4.
Show (and the previous problem may be helpful) that the e.g.f. is
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This note was uploaded on 05/19/2010 for the course MATH mat 146 taught by Professor Gregkuperberg during the Spring '10 term at UC Davis.
 Spring '10
 GregKuperberg

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