MAT 280 — UC Davis, Winter 2011
Longest increasing subsequences and combinatorial probability
Homework Set 1
Homework due: Wednesday 2/2/11
1. Let Λ = lim
n
→∞
‘
n
/
√
n
as in Theorem 6 in the lecture notes. The goal of this
problem is to show that the bounds 1
≤
Λ
≤
e
that follow from Lemmas 3
and 4 can be improved to (8
/π
)
1
/
2
≤
Λ
≤
2
.
49.
(a) In the proof of Lemma 4 observe that if
L
(
σ
n
)
≥
t
then
X
n,k
≥
(
t
k
)
, so the
bound in (1.4) in the notes can be improved. Take
k
≈
α
√
n
and
t
≈
β
√
n
and optimize the improved bound over
α < β
(using some version of Stirling’s
formula) to conclude that Λ
≤
2
.
49.
(b) Given a standard Poisson Point Process (PPP) Π in [0
,
∞
)
×
[0
,
∞
), con
struct an increasing subsequence (
X
1
,Y
1
)
,
(
X
2
,Y
2
)
,
(
X
3
,Y
3
)
,...
of points from
the process by letting (
X
1
,Y
1
) be the Poisson point that minimizes the coordi
nate sum
x
+
y
, and then by inductively letting (
X
k
,Y
k
) be the Poisson point in
(
X
k

1
,
∞
)
×
(
Y
k

1
,
∞
) that minimizes the coordinate sum
x
+
y
. Observe that
the properties of the Poisson point process imply that one can write
(
X
k
,Y
k
) =
k
X
j
=1
(
W
j
,Z
j
)
,
where
(
(
W
j
,Z
j
)
)
∞
j
=1
is a sequence of independent and identically distributed
random vectors in [0
,
∞
)
2
, each having the same distribution as (
X
1
,Y
1
). Com
pute the expectations
μ
X
=
E
(
X
1
)
,μ
Y
=
E
(
Y
1
)
.
Hint: