Combinatorics Through Guided Discovery, 2004a

20 1. What is Combinatorics?
a Proving this takes more of a detour than is advisable here; however there is an elementary
proof which you can work through in the problems of the end of Section 1 of Chapter 1 of
Introductory Combinatorics by Kenneth P. Bogart, Harcourt Academic Press, (2000).
1.3 Some Applications of the Basic Principles
1.3.1 Lattice paths and Catalan Numbers
◦ Problem 47. In a part of a city, all streets run either north-south or east-west,
and there are no dead ends. Suppose we are standing on a street corner. In
how many ways may we walk to a corner that is four blocks north and six
blocks east, using as few blocks as possible? (h)
· Problem 48. Problem 47 has a geometric interpretation in a coordinate
plane. A lattice path in the plane is a “curve” made up of line segments
that either go from a point (i, j) to the point (i +1, j) or from a point (i, j)
to the point (i, j +1), where i and j are integers. (Thus lattice paths always
move either up or to the right.) The length of the path is the number of such
line segments.
(a) What is the length of a lattice path from (0, 0) to (m, n)?
(b) How many such lattice paths of that length are there? (h)
(c) How many lattice paths are there from (i, j) to (m, n), assuming i, j,
m, and n are integers? (h)
· Problem 49. Another kind of geometric path in the plane is a diagonal
lattice path. Such a path is a path made up of line segments that go from
a point (i, j) to (i +1, j +1)(this is often called an upstep) or(i +1, j − 1)
(this is often called a downstep), again where i and j are integers. (Thus
diagonal lattice paths always move towards the right but may move up or
down.)
(a) Describe which points are connected to (0, 0) by diagonal lattice
paths. (h)
(b) What is the length of a diagonal lattice path from (0, 0) to (m, n)?
(c) Assuming that (m, n) is such a point, how many diagonal lattice paths
are there from (0, 0) to (m, n)? (h)