Combinatorics Study Guide

Chapter 3: Subsets, Partitions, PermutationsThe number of subsets is 2n

The binomial coe�cient�nk

�is the number of subsets of size k from a set of n elements. Binomial coe�cient identities:�

nk

�= n!

k!(n�k)!�nk

�=�

nn�k

�k�nk

�= n

�n�1k�1

��n+1k

�=�

nk�1

�+�nk

� Pnk=0

�nk

�= 2n

Pnk=0

�nk

�2=�2nn

�Thm: \Binomial Theorem" (1 + t)n =

Pnk=0

�nk

�tk. *Strong form: (x+ y)n =

Pnk=0

�nk

�xkyn�k.

Fact: For n > 0, the numbers of subsets of an n-set of even and of odd cardinality are equal.P

0�k�n

k odd

�nk

�=P

0�k�n

k even

�nk

�

Prop: If n is a multiple of 8, then the number of sets of size divisible by 4 is 2n�2 + 2(n�2)=2.Thm: \Lucas' Theorem" Let p be prime, and let m = a0+a1p+ � � �+akp

k; n = b0+b1p+ � � �+bkpk, where 0 � ai; bi < p

for i = 0; : : : ; k � 1. Then�mn

� �Qki=0

�aibi

�(mod p).

Def:A permutation of a set X is a one-to-one mapping from X to itself.Prop: The number of permutations of an n-set is n!Prop:Any permutation can be written as the composition of cycles on pairwise disjoint subsets. the representation is

unique, apart from the order of the factors, and the starting-points of the cycles.Thm: \Stirling's Formula" n! =

p2�n

�ne

�n �1 +O

�1n

��:

Prop: 22n=(2n+ 1) � �2nn

� � 22n

Lma: The number of n-tuples of non-negative integers x1; : : : ; xn with x1 + � � �+ xn = k is�n+k�1n�1

�=�n+k�1

k

�.

Lma: The number of choices of k objects from n with repetitions allowed and order not signi�cant is equal to the numberof ways of choosing n non-negative integers whose sum is k.

Thm: The number of selections of k objects from a set of n objects are given by the following table:Order Signi�cant Order Not Signi�cant

Repetitions allowed nk�n+k�1

k

�Repetitions not allowed n(n� 1) � � � (n� k + 1) = n!

k!

�nk

�Prop: The number of ordered selections without repetition from a set of n objects is be � n!c, where e is the base of

natural logarithms.Def:A relation R on X is a subset of X2. R is re exive if 8x 2 X; (x; x) 2 R. R is irre exive if 8x 2 X; (x; x) =2 R.

R is symmetric if 8x; y 2 X; (x; y) 2 R implies (y; x) 2 R. R is antisymmetric if whenever (x; y) 2 R and(y; x) 2 R, then x = y. R is transitive if, for all x; y; z 2 X; (x; y) 2 R and (y; z) 2 R together imply (x; z) 2 R.

Def:An equivalence relation is a re exive, symmetric, and transitive relation. For x 2 X and an equivalence relationR, the equivalence class containing x is the set R(x) = fy 2 Xj(x; y) 2 Rg. A partition of X is a family ofpairwise disjoint non-empty subsets whose union is X.

Thm: Let R be an equivalence relation on X. Then the equivalence classes of R form a partition of X. Conversely,given any partition of X, there is a unique equivalence relation on X whose equivalence classes are the parts of thepartition.

Def:A relation R on X is a partial order if it is re exive, antisymmetric, and transitive. A relation R is said tosatisfy trichotomy if for any x; y 2 X, one of the cases (x; y) 2 R; x = y; or (y; x) 2 R holds. R is a total order,or order, if it is a partial order which satis�es trichotomy.

Def:A relation R is a partial preorder, or pre-partial order, if it is re exive and transitive. A parial preordersatisfying trichotomy is a preorder.

[Skipping Section 3.9: Finite Topologies]Thm: \Cayley's Theorem on Trees" The number of labeled trees on n vertices is nn�2.Def:A vertebrate is a tree with two distinguished vertices called the head and tail. An endofunction on N is a

function form N to itself.Prop: The number of vertebrates and endofunctions on N are equal.Def: The Bell Number Bn is the number of partitions of an n-set.Thm: \Recurrence for Bell Numbers" B0 = 1. For n � 1, Bn =

Pnk=1

�n�1k�1

�Bn�k.

Chapter 4: Recurrence relations and generating functionsDef:A recurrence relations expresses the value of a function f at the natural number n in terms of its values at

smaller natural numbers. A (k + 1)-term recurrence relation expresses any value F (n) of a function in termsof the k preceding values F (n � 1); F (n � 2); : : : ; F (n � k); it is linear if it has the form F (n) = a1(n)F (n � 1) +a2(n)F (n� 2) + � � �+ ak(n)F (n� k) where each ai is a function of n; and it is linear with constant coe�cients

if each ai is constant. Generating functions are power series whose coe�cients from the number sequence inquestion.

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Combinatorics Study Guide

Fact:A function satisfying a (k+1)-term recurrence relation is uniquely determined by its values on the �rst k naturalnumbers.

Def (Fibonacci Recurrence Relation): For n � 2 Fn = Fn�1 + Fn�2.

Fact: The Fibonacci numbers are exactly those of the form Fn =�p

5+12p5

��1+p5

2

�n+�p

5�12p5

��1�p5

2

�n.

Def: The characteristic equation of a linear recurrence relation is the polynomial formed by simplifying the recurrencerelation for a solution of the form �n. If all roots are distinct, then there are k independent solutions of the recurrencerelation, which can be linearly combined for the full solution.

Def:A derangement of 1; 2; : : : ; n is a permutation of this set which leaves no point �xed. Its recurrence relation isDn = (n� 1)(Dn�1 +Dn�2).

Thm: The number Dn of derangements of an n-set is given by Dn = n!�Pn

i=0(�1)ii!

�. This is the nearest integer to

n!=e.Def: Let s(n) denote the number of permutations of a set of n elements that partitions into cycles of length at most 2.

This follows the relation s(n) = s(n� 1) + (n� 1)s(n� 2).Prop:A. s(n) is even for all n > 1. B. s(n) >

pn! for all n > 1.

Note: The evenness of s(n) relates to thisFact: in a �nite group G, the number of solutions of x2 = 1 is even.Section 4.5: Catalan and Bell Numbers

Def: The Catalan numbers are the number of ways a sum of n terms can be bracketed so that it can be calculatedby adding two terms at a time. Its recurrence relation is Cn =

Pn�1i=1 CiCn�i. It's direct form is Cn = 1

n

�2n�2n�1

�.

Prop:An exponential generating form for the Bell numbers isP

n�0Bnt

n

n! = exp(exp(t)� 1).

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