Math 146: Algebraic Combinatorics
Homework 8
This problem set is due Friday, May 21. This is a half problem set.
Do the following problems:
GK8.1.
An
alternating permutation
is a permutation
σ
of
[
n
]
such that
σ
(
k
+
1
)

σ
(
k
)
is positive when
k
is odd and negative when
k
is even. For example, 1423 is an alternating permutation. Let
a
n
be the
number of alternating permutations of
[
n
]
, and let
A
(
x
)
be their exponential generating function.
(a)
Let
b
n
be the set of “zigzag permutations”, which are those that are either alternating or satisfy
the reverse condition, that
σ
(
k
+
1
)

σ
(
k
)
is positive when
k
is even and negative when
k
is
odd. Show that
b
n
=
2
a
n
and thus
B
(
x
) =
2
A
(
x
)
for their generating functions. (We set
b
0
=
2
by default.)
*(b)
Prove the “two animal” relation
B
0
(
x
) =
A
(
x
)
2
+
1
.
(c)
Use (a) and (b) and the initial condition
A
(
0
) =
a
0
=
1 to establish that
A
(
x
) = (
sec
x
)+(
tan
x
)
.
GK8.2.
The version of Abel’s theorem that I gave in class, although correct, is not the strongest one avail
able for our purposes. It is a theorem that if
f
(
x
)
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 Spring '10
 GregKuperberg
 Permutations, Generating function

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