1. Mathematics for Computer Science MIT 6.042J/18.062J
Combinatorics I
Copyright © Radhika Nagpal, 2002.
Prof. Albert Meyer & Dr. Radhika Nagpal

2. Counting 4-digit Strings with 7s
All strings minus strings with no 7s
Count strings based on the first occurrence of 7
7xxx + o7xx + oo7x + ooo7
Count strings based on how many 7s are there
strings with one 7 + strings with two 7s + strings with three 7s + strings with four 7s

3. Counting Techniques Bijections Sum Rule, Inclusion-Exclusion
Product Rule
Pigeonhole Principle
Trees
Permutations

4. Counting Strings (again)
How many strings of length 3 can make with the alphabet A = {r,a,d} ?

5. Counting Strings (3) (3) (3) {r,a,d} {r,a,d} {r,a,d}
(3) (3) (3)
{r,a,d} {r,a,d} {r,a,d}
Number of strings = = 27

6. Counting Rearrangements
How many strings of length 3 can I make by rearranging the letters rad?

7. Permutations (3) (2) (1) {r,a,d} {?,?} {?}
(3) (2) (1)
{r,a,d} {?,?} {?}
Number of strings, |S| = = 6

8. Permutations (3) (2) (1) {r,a,d} {?,?} {?} S = {rad, rda ard, adr
(3) (2) (1)
{r,a,d} {?,?} {?}
S = {rad, rda
ard, adr
dra, dar}

9. Permutation Tree d
rad
rda
ard
adr
dra
dar a a d d r r a d r d r a a r

10. … … Strings Tree r a d r r a a d d r r a a d d rrr rra rrd rar raa rad
rda
rdd
…..
drr
dra
drd
dar
daa
dad
ddr
dda
ddd a d r r a a d d r r … a a d … d

11. Comparison rearrangement strings n n2 n! nn 1 1 1 1 5 25 120 3125 ~ ~ ~ ~
scheme dies…..

12. Important technique: Matching
Make a one-to-one mapping between a given problem and a set of known cardinality (or at least one we know how to count…)
Functions Binary strings
Lists Trees
Graphs, Relations Matrices
Strings, Polynomials Balls and Bins

13. Counting Functions f B A
How many different possible functions are there from A to B? f b1 b2 b3 b4 b5 b6 a1 a2 a3 a4 a5 A B

14. Mapping: Functions as Strings
a1 a2 a3 a4 a5 A B
b1 b1 b1 b1 b1
b1 b2 b3 b1 b4 ……
string of length 5 from
the alphabet {b1, b2…b6} f

15. Counting Functions A B (6) (6) (6) (6) (6)
a1 a2 a3 a4 a5 A B
(6) (6) (6) (6) (6)
Total number of possible functions
= total number of possible strings
= =

16. Counting Bijections f B A
How many different possible bijections are there from A to B? f a1 a2 a3 a4 a5 b1 b2 b3 b4 b5 A B

17. Counting Bijections A B Permutations of the letters {b1,b2,b3,b4,b5}
a1 a2 a3 a4 a5 A B
b1 b2 b3 b4 b5
b5 b4 b3 b2 b1 ……
Permutations of the
letters {b1,b2,b3,b4,b5}

18. Counting Bijections A B (5) (4) (3) (2) (1)
a1 a2 a3 a4 a5 A B
(5) (4) (3) (2) (1)
Total number of possible functions
= total number of possible permutations
= = 5!

19. Counting Graphs v1 v1 v1 v2 v3 v2 v3 v2 v3
How many possible directed graphs are there on n nodes? v1 v1 v1 v2 v3 v2 v3 v2 v3

20. Graphs as Binary Strings
Maximum possible edges = n2
e1 e2 e3 …… en2
G ….. 1
G ……1 G

21. Graphs as Boolean Matrices
0 1 1 2 3 . n
Any assignment
of 1s and 0s is a possible graph

22. Proving that a Mapping is a Bijection
Prove that g: F S is a bijection
Prove that all elements in F map to a unique elements of S
Prove that all elements in S map to some F

23. Example: Graph Isomorphism
G1 and G2 are isomorphic if there is a bijection f from the nodes of G1 to the nodes of G2, that preserve the adjacency lists.
Question: At most how many functions do I need to check to determine whether or not G1 and G2 are isomorphic?
Answer: All possible bijections from the nodes of G1 to G2 = n!

24. In-class Problems