CRN: 87401
MAD 6206 Combinatorics I, Fall 2011
Assignment 2 suggested solution
Question 1. (A recurrence for p(k, n)
Recall that p(k, n) equals the number of partitions of k into n
parts. We adopt the convention that p(k, 0) = p(0, k ) = k,0 . Also, p(k )

CRN: 87401
MAD 6206 Combinatorics I, Fall 2011
On Gospers Algorithm
The aim is to nd a simple form for the sum S (m) = m tn where r(n) = tn+1 /tn is a rational function
n=0
of n. That is, tn is a hypergeometric term in n and does not depend on m.
1
A Brie

CRN: 87401
MAD 6206 Combinatorics I, Fall 2011
Brief notes/example on hypergeometric series
1
Denition
Consider the series k0 tk where tk can depend on some other parameters (and such parameters are
regarded as constants). If
tk+1
= x,
for some constant x

MAD 6206 Combinatorics I, Fall 2011
MWF 12:55-1:45pm, PHY 120
Instructor:
Oce:
Email:
Oce hours:
CRN #: 87401
Stephen Suen
PHY356
ssuen@usf.edu
T 9-10am, R 8:30-10:30am (to be conrmed).
Course Description We shall consider enumeration techniques in this c

CRN: 87401
MAD 6206 Combinatorics I, Fall 2011
Assignment 7 suggested solution
Question 1. (Rotations of the 3-d cube) We talked about coloring the 6 faces of the cube in class so that
each face has a dierent color. We shall color the vertices and edges o

CRN: 87401
MAD 6206 Combinatorics I, Fall 2011
Assignment 7
Question 1. (Rotations of the 3-d cube) We talked about coloring the 6 faces of the cube in class so that
each face has a dierent color. We shall color the vertices and edges of the cube in this

CRN: 87401
MAD 6206 Combinatorics I, Fall 2011
Assignment 6 suggested solution
Question 1. (Two well-known series in hypergeometric notation) (a) Consider the sum k n+k z k . We
k
know how to nd a closed form from other methods. In particular, the sum fol

CRN: 87401
MAD 6206 Combinatorics I, Fall 2011
Assignment 5 suggested solution
Question 1. (Eulerian numbers and their generating function) Recall that the the Eulerian numbers
(A(n, k ) = An,k ) are dened as the number of permutations on [n] with k runs.

CRN: 87401
MAD 6206 Combinatorics I, Fall 2011
Assignment 5
Question 1. (Eulerian numbers and their generating function) Recall that the the Eulerian numbers
(A(n, k ) = An,k ) are dened as the number of permutations on [n] with k runs. We proved in class

CRN: 87401
MAD 6206 Combinatorics I, Fall 2011
Assignment 4 suggested solution
Question 1. (Catalan numbers and sequences of +1s and 1s) Assume that p, r are nonnegative
integers. Let p,r be the number of sequences cfw_ai i1 , where ai = 1, with p 1s and

CRN: 87401
MAD 6206 Combinatorics I, Fall 2011
Assignment 4
Question 1. (Catalan numbers and sequences of +1s and 1s) Assume that p, r are nonnegative
integers. Let p,r be the number of sequences cfw_ai i1 , where ai = 1, with p 1s and p + r +1s (so
2p+r

CRN: 87401
MAD 6206 Combinatorics I, Fall 2011
Assignment 3 suggested solution
Question 1. (Binary strings and Fibonacci numbers) Suppose that the digit 0 takes one unit of time
to transmit through a communication channel while the digit 1 takes two units

CRN: 87401
MAD 6206 Combinatorics I, Fall 2011
Assignment 3
Question 1. (Binary strings and Fibonacci numbers) Suppose that the digit 0 takes one unit of time
to transmit through a communication channel while the digit 1 takes two units of time. (Then the

CRN: 87401
MAD 6206 Combinatorics I, Fall 2011
Assignment 1 - suggested solution
Question 1. (Multinomial coecients) Let 1 and n, k1 , k2 , . . . , k be nonnegative integers satisfying
k1 + k2 + + k = n.
Dene the multinomial coecient
n
k1 ,k2 ,.,k
as the