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MAS 219 Combinatorics

Assignment 4 For handing in on 24 October 2007

Write your name and student number at the top of your assignment before

handing it in. Staple all the pages together. Post the assignment in theRED post-box on the 2nd floor of the Maths building before 16:30 on

Wednesday.

1 You are standing at the point (0,0) in the plane, and you wish to reach the point(2n,0). In one move you are allowed to go from (x,y) to either (x + 1,y + 1) or(x + 1,y1): that is, one step in the north-east or south-east direction. However, youare not allowed to visit any point below the X-axis, that is, all points on the path have

non-negative y-coordinate. The two possible paths for n = 2 are shown in the picture.

r r r r r r r r r r

r r r r r r r r r r

r r r r r r r r r r

(0,0) (0,0)(4,0) (4,0)

ddd

dd

d

d

dd

ddd

Let F(n) be the number of such walks, and let G(n) be the number of walks whichdo not touch the X-axis except at the start and end. The picture shows F(2) = 2 andG(2) = 1.

(a) Show by drawing diagrams that F(3) = 5 and G(3) = 2.

(b) By considering walks which after leaving the origin first touch the X-axis at the

point (2k,0), show that

F(n) =n

k=1

G(k)F(n k),

where by convention we take F(0) = 1. [Hint: The first 2k steps take us from(0,0) to (2k,0) without touching the axis except at the start and end; the last2(n k) steps can be regarded as a walk from (0,0) to (2(n k),0), shiftedright 2k places.]

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(c) Show that G(n) = F(n1). [Hint: Take any walk with 2(n1) steps counted

by F(n1), say W. Now do the following. Start at (0,0). Take one north-eaststep, reaching (1,1); then follow the steps of the walkW, ending at (2n1,1);then take one south-east step, to (2n,0). Show that this walk is one of thosecounted by G(n), and moreover every such walk arises in this way.]

(d) Hence show that F(n) = Cn+1, the (n + 1)st Catalan number, for n 0. [Hint:Induction.]

2 (a) Prove each of the following statements (i) by directly counting the partitions,

(ii) by using the recurrence relation:

S(n,2) = 2n11 for n 2;

S(n,n1) =

n

2

for n 2.

(b) Find and prove a formula for S(n,n2) for n 3.

The following question is not for credit but I will be happy to look at and comment

on solutions.

3 Let dn be the number of derangements of{1,2, . . . ,n}.

(a) Prove that dn = ndn1 + (1)n for n 1.

(b) Deduce the formula dn = n!

n

k=0

(1)k

k!

for n 0.

(c) Let D(x) = n0

dnxn

n!. Prove that D(x) =

ex

1x.

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