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

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

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

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

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 = 3.3.3 = 27

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

Permutations (3) (2) (1) {r,a,d} {?,?} {?} (3) (2) (1) {r,a,d} {?,?} {?} Number of strings, |S| = 3.2.1 = 6

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}

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

… … 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

Comparison rearrangement strings n n2 n! nn 1 5 10 20 100 1000

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

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

Mapping: Functions as Strings a1 a2 a3 a4 a5 A B

Mapping: Functions as Strings a1 a2 a3 a4 a5 A B b1 b1 b1 b1 b1 b1 b2 b3 b1 b4 ……

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

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 = 6.6.6.6.6 =

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

Counting Bijections A B a1 a2 a3 a4 a5 b1 b2 b3 b4 b5 b5 b4 b3 b2 b1 ……

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}

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 = 5.4.3.2.1 = 5!

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

Graphs as Binary Strings Maximum possible edges = e1 e2 e3 ………e G1 1 0 1 0 ….. 1 G2 0 1 1 0……1 G3 ..

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

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

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.

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?

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!

In-class Problems