Notation Throughout the book the following notation is used: a A The element a belongs to the set A A B A is a subset of B {a A ψ(a)} The elements of A that satisfy property ψ {a 1,a 2,...,a r } The set consisting of the elements a 1,a 2,...,a r A B The intersection of the sets A and B A B The union of the sets A and B P(A) The power set of A C The intersection of all sets of C C The union of all sets of C The empty set A The number of elements of set A y The absolute value of the real number y x<y xis smaller than y x y x is smaller than or equal to y x y x is not equal to y x The smallest integer greater than or equal to x x The greatest integer smaller than or equal to x a b(mod c) a is congruent to b modulo c, a b is divisible by c j k=i ψ(k) ψ(i)+ ψ(i + 1) + ψ(i + 2) + +ψ(j) j k=i ψ(k) ψ(i) ψ(i + 1) ψ(i + 2)... ψ(j) n! ( 1 2 3... n n k) F n L n c n s(n,k) c(n,k) S(n,k) r(l,s) m n v(g) e(g) n! k!(n k)! The Fibonacci numbers The Lucas numbers The Catalan numbers The Stirling numbers of the first kind The unsigned Stirling numbers of the first kind The Stirling numbers of the second kind The Ramsey numbers The dimensions of a board of m rows and n columns The set of vertices of a graph G The set of edges of a graph G 169
170 Notation d(v) The degree of vertex v N v The connected component of v Γ(S) The vertices adjacent to some vertex of S f : A B f is a function from A to B f [A] The image of the set A under f f(a) The element assigned to a by f σ τ The composition of σ with τ σ 1 The inverse permutation of σ σ A σ restricted to A (defined only if σ [A]=A) (γ 1,γ 2,...,γ k ) The permutation that sends each γ i to γ i+1 and γ k to γ 1 (a 0,a 1,a 2,...) The sequence a 0,a 1,... f (x) The derivative of the generating function f (t) n t(t 1)...(t n + 1),ifn 1, (t) 0 = 1 When making references to problems the following abbreviations were used IMO International Mathematical Olympiad OIM Iberoamerican Mathematical Olympiad APMO Asian Pacific Mathematical Olympiad OMCC Centroamerica and the Caribbean Mathematical Olympiad OMM Mexican Mathematical Olympiad USAMO USA Mathematical Olympiad (Country, year) The problem was used in the olympiad of that country in one of its stages and the corresponding year
Further Reading 1. Pérez Seguí, M. L., Combinatoria, Cuadernos de Olimpiadas de Matemáticas. Instituto de Matemáticas, UNAM, 2000. 2. Riordan, J., Introduction to Combinatorial Analysis. Dover, Mineola, 2002. 3. Andreescu, T. and Feng, Z., A Path to Combinatorics for Undergraduates. Birkhäuser, Basel, 2004. 4. Andreescu, T. and Feng, Z., 102 Combinatorial Problems. Birkhäuser, Boston, 2003. 5. Anderson, I., A First Course in Combinatorial Mathematics, 2nd edition. Oxford University Press, London, 1989. 6. Stanley, R. P., Enumerative Combinatorics, 2nd edition, Vol. 1. Cambridge University Press, Cambridge, 2011. 171
Index A Absolute convergence theorem, 88 B Bijective function, 59, 69 Binet s formula, 82 Binomial coefficients, 3 extended, 79 C Cardinality of a set, 2 Catalan numbers, 82 85 Center of gravity, 18 Clique of a graph, 47 Closed formula, 78 Coloring, 31 Commuting permutations, 66 Complete graph, 47 Composition of functions, 59 Conjugate partition, 94 Connected component, 49 Convergence, 88 Cycle, 65 Cycle decomposition, 66 Cycle structure (of a permutation), 75 D Degree, 44 Delimiters, 12 Derivative, 85 87 Diophantine equation, 12 Dirichlet principle, 17 Disjoint cycles, 65 Distance in a graph, 49 E Edge of a graph, 43 Empty graph, 43 Empty set, 1 Equilibrium, 35 Erdős-Ko-Rado theorem, 72 Erdős-Szekeres theorem, 22 Euler, 43 Euler constant, 14 Extended binomial coefficients, 79 F Factorial, 3 Fermat s theorem, 23 Ferrer diagram, 93 Fibonacci numbers, 7, 24, 80 82, 90 Fixed point, 64 Forest (graph), 49 Function, 59 72 generating, 77 90 G Game theory, 35 Gauss formula, 5 Generating function, 77 90 product, 78 sum, 78 Golden ratio, 80 Graph, 43 55, 66, 69 bipartite, 51 53 connected, 47 connected component, 49 degree, 44 edge, 43 independent, 51 matching, 53 55 perfect, 53 of a board, 52 simple, 44 spanning tree, 48 subgraph, 43 tree, 47 triangle inequality, 50 vertex, vertices, 43 walk, path, cycle, 45 173
174 Index H Hall s theorem, Marriage theorem, 54 Hamiltonian cycle, 138 I Image of a function, 59 Incidence, 44 Inclusion-exclusion principle, 13 Independent term, 78 Induction, 5 9 Infinite descent, 29 Infinite pigeonhole principle, 19 Injective function, 59 Intersection of sets, 1 Invariant, 27 Inverse of a permutation, 63 K König, 43 König s theorem, 51 L Law of the product, 2 Law of the sum, 2 Losing position, 34 Lucas numbers, 90 M Mantel s theorem, 55 Matrix, 68 Möbius function, 154 Möbius inversion formula, 154 Multigraph, 44 Multiset, 43 N Neighborhood, 54 Newton s theorem, 3 O Orbit, 64 Order of an element (in a permutation), 64 P Partial fraction decomposition, 82 Partition, 93, 94, 96 98 conjugate, 94 Ferrer diagram, 93 of a set, 93 of an integer, 93 self-conjugate, 94 Pascal, 6 Pascal s formula, 4 Paths in boards, 9 Permutation, 3, 63 69 orbit, 64 with k cycles, 95 without fixed points, 14 Pigeonhole principle, 17 25 Power set, 2 R Radius of a graph, 145 Ramsey numbers, 20, 21 Recursive equation, 77 Recursive relation, 77 Russell s paradox, 1 S Second moment, 30 Self-conjugate partition, 94 Set, 1 Simple graph, 44 Spanning tree, 48 Stirling numbers, 94 98 of the first kind, 95 of the second kind, 96 unsigned (first kind), 96 Strong induction, 7 Subgraph, 43 Subset, 1 Surjective function, 59 System of representatives, 57 T Taylor series, 89 Transposition, 65 Tree, 47 Triangle inequality, 50 U Union of sets, 1 Unsigned Stirling numbers of the first kind, 96 V Vandermonde s formula, 11 Vertex of a graph, 43 W Wagner graph, 123 Walk, 45 Winning position, 34 Winning strategy, 33