MAT5107 : Combinatorial Enumeration
Mike Newman, 17 February
Assignment 2
turn page over, there are 7 questions
P
1. Let an , bn for n ≥ 0 be two infinite sequences related by bn = k nk ak . Find an expression for an in
terms of some of the bk .
P
(hint: Think of k nk ak as being a term in the product of two GFs (ordinary or exponential, as you
see fit). Identify those GFs, and obtain an equation relating the GFs for an and bn .)
√
1 − 4x. Verify that it is consistent with the binomial theorem
√ applied to
1 − 1 − 4x
(1 − y) with y = 4x and n = 1/2. Using this series, find the series expansion for
. (as a
2x
hint, you should already know the final answer)
2. Find the Taylor series of
n
3.
4.
a)
Let R be the set of rooted plane trees where the number of children
P at any vertex is constrained
to be a multiple of 5. Derive a functional equation for R(x) = m≥0 rm xm (similar to the one
we had for rooted plane trees in class), where rm is the number of such rooted plane trees with
m edges.
Note: The number of children does not have to be the same at each vertex, it just has to be a
multiple of 5.
b)
Show that your functional equation has a unique solution as a formal power series. That is, show
that the functional equation uniquely determine all of the rm . You do not need to find an explicit
formula for the rm .
a)
Let D be any subset (possibly infinite) of non-negative integers. Let T be the set of all rooted
plane trees such that P
at each vertex the number of children is an element of D. Derive a functional
equation for T (x) = m≥0 tm xm .
b)
Show that this equation (and maybe some initial values) admits a unique solution for T (x). You
do not need to find T explicitly, just explain how all of the tm are uniquely determined by this
equation.
c)
Show that if 0 ∈
/ D then you can find T explicitly directly from your functional equation (and
maybe a few initial values). Explain how you could have deduced, without any generating functions, what the tm are, and hence what T (x) is in the case where 0 ∈
/ D.
5. Suppose that A(x) is an EGF that satisfies a linear homogeneous differential equation. That is
A(x) =
X
n≥0
xn
an
n!
and
0=
r
X
t=0
ct
dt
A(x)
dxt
Show that an satisfies a linear homogeneous recurrence. What, exactly, is the recurrence? What initial
values do we need to know before we can use the recurrence?
6. Let d ≥ 1, and define S(x) =
a)
X
n
X
n≥0
k=0
!
k
d
xn . So sn = [xn ] S(x) = 0d + 1d + 2d + · · · + nd .
Find an expression for S(x). You should take inspiration from the equation preceding Probd
lem 4.25 in the notes; your equation can involve dx
etc.
b)
Prove that S(x) is of the form
d
X
j=1
cj xj
, where the cj are positive constants. You don’t
(1 − x)j+2
need to determine the cj precisely.
(hint: you might think of showing this by induction on d, and you might think of using the
product rule instead of the quotient rule).
c)
By computing [xn ] S(x) from the expression given above, give sn in terms of the cj and binomial
coefficients. Based on this, show that sn is a polynomial in n of degree d + 1. You don’t need to
determine this polynomial.
7. You might “recall” that the ordinary generating function for an = 1/n is A(x) =
X xn
1
= log
.
n
1−x
n≥1
Z
X xn
1
=
.
In case you didn’t “recall” this you might observe that
n
1−x
n≥1
In the following you should use corollary 7.2 to obtain the generating functions. You can observe
after that fact that that there are other ways to obtain the generating function, but you should use
Corollary 7.2.
a)
Give a closed form for
X x5n
. Your answer should simplify quite a bit (involving no imaginary
5n
n≥1
numbers and perhaps only a single occurrence of “x”?), but this should not need a lot of work.
X x2n−1
b) Give a closed form for
. Your answer should simplify quite a bit (involving no imaginary
2n − 1
n≥1
numbers).
c)
Give a closed form for
X x5n+2
. Your answer can be in terms of imaginary numbers; you don’t
5n + 2
n≥0
need to simplify.