W 10a
Bijective Proof Requirements: For n 0, let A n and B n be sets of
combinatorial objects. Set an := |An| and bn := |Bn|. If you are asked to
provide a bijective proof that an = bn for n 0, you are to do the following
for general n 0: Define a functio
W9
(7.48) For each of Parts (a) and (g), do only Steps (i) and (ii) below: Find
the generating function H(x) for the sequence determined by the recurrence and
the initial conditions, as on p. 235. (We will not discuss the partial fractions
technique, whic
W8
(7.37)(a) Find a recurrence for n 2. Hint: Without the special restriction,
for strings of length 2 or more there would (normally) be nine possibilities for
the last 2 entries: 00, 01, , 22. Here the endings 00 and 11 are being
disallowed. This leaves
W6
Chapter 6 Exercises: Exercises 11 - 16 are interrelated (especially 12, 14,
16), and so you should give all of them a first try (reading their hints/answers)
before spending too much time being stuck on any one of them in isolation.
(Exercises 12, 15,
W5
Bijective Proof That Two Sets Have the Same Cardinality
Let S and T be two finite sets, each containing its own kind of combinatorial
objects. Often in this course we will show that |S| = |T| by describing a one-toone correspondence (bijection) between
W4
Confusions with Distribution Problems: Since using our selection counts
Theorems 2.4.1, 2.2.1, 2.3.1, 2.5.1 to solve distribution problems is the trickiest
part of this course for most students, please read this paragraph and the
statement for Problem
C h a p t er s 11, 12, 13:
G r a p h T h eor y
At the beginning of the semester it was announced that this course would be
primarily concerned with the oldest and most renowned part of combinatorics,
the enumeration of combinatorial objects. But in additi
W 10b/11a
( V 3 41/2) Consider the bipartite graph G = (X,Y) in which
X = cfw_1,2,3,8,9, Y = cfw_a,b,c,h,i, and has many edges. In particular,
the edge set contains both the matching
M = cfw_(1,a), (3,c), (5,i), (6,f), (7,d), (8,e), (9,h) and the matching
Homework #8 Spring 2012
(7.37)(a) Let n 2 2. Consider the set of such strings of length 11. None of
these end with 00 or ii. There are an_2 of these which end with 22. The
remaining strings each end with one of the six possibilities: 01, 02, 10, 12, 20,