\contentsline {part}{I\hspace {1em}Fundamentals}{1}{part.1}%
\contentsline {chapter}{\numberline {1}Basic combinatorial methods}{5}{chapter.1}%
\contentsline {section}{\numberline {1.1}Bijection principle}{6}{section.1.1}%
\contentsline {section}{\numberline {1.2}Pidgeonhole principle}{6}{section.1.2}%
\contentsline {section}{\numberline {1.3}Sum and product rules}{8}{section.1.3}%
\contentsline {subsection}{\numberline {1.3.1}Sum rule}{8}{subsection.1.3.1}%
\contentsline {subsection}{\numberline {1.3.2}Product rule}{9}{subsection.1.3.2}%
\contentsline {subsection}{\numberline {1.3.3}Quotient rule}{9}{subsection.1.3.3}%
\contentsline {section}{\numberline {1.4}Factorial function}{10}{section.1.4}%
\contentsline {section}{\numberline {1.5}Recursive formulae}{11}{section.1.5}%
\contentsline {subsection}{\numberline {1.5.1}Fibonacci numbers}{11}{subsection.1.5.1}%
\contentsline {chapter}{\numberline {2}Arrangements, permutations, and combinations}{13}{chapter.2}%
\contentsline {section}{\numberline {2.1}Arrangements}{14}{section.2.1}%
\contentsline {subsection}{\numberline {2.1.1}String notation}{14}{subsection.2.1.1}%
\contentsline {section}{\numberline {2.2}Permutations}{15}{section.2.2}%
\contentsline {section}{\numberline {2.3}Combinations}{16}{section.2.3}%
\contentsline {section}{\numberline {2.4}Binomial coefficient}{18}{section.2.4}%
\contentsline {chapter}{\numberline {3}Partitions}{19}{chapter.3}%
\contentsline {section}{\numberline {3.1}Integer partitions}{19}{section.3.1}%
\contentsline {subsection}{\numberline {3.1.1}Integer partition function}{19}{subsection.3.1.1}%
\contentsline {subsection}{\numberline {3.1.2}Young diagram}{19}{subsection.3.1.2}%
\contentsline {section}{\numberline {3.2}Set partitions}{20}{section.3.2}%
\contentsline {subsection}{\numberline {3.2.1}Bell numbers}{20}{subsection.3.2.1}%
\contentsline {subsection}{\numberline {3.2.2}Stirling numbers}{21}{subsection.3.2.2}%
\contentsline {chapter}{\numberline {4}Further elementary methods}{23}{chapter.4}%
\contentsline {section}{\numberline {4.1}Twelvefold way}{23}{section.4.1}%
\contentsline {section}{\numberline {4.2}Generating polynomials}{23}{section.4.2}%
\contentsline {section}{\numberline {4.3}Inclusion-exclusion principle}{23}{section.4.3}%
\contentsline {subsection}{\numberline {4.3.1}Derangement numbers}{24}{subsection.4.3.1}%
\contentsline {section}{\numberline {4.4}Catalan numbers}{26}{section.4.4}%
\contentsline {chapter}{\numberline {5}Combinatorial numbers}{27}{chapter.5}%
\contentsline {section}{\numberline {5.1}Fibonacci sequence}{27}{section.5.1}%
\contentsline {subsection}{\numberline {5.1.1}Sums of 1 and 2}{28}{subsection.5.1.1}%
\contentsline {section}{\numberline {5.2}Catalan numbers}{28}{section.5.2}%
\contentsline {subsection}{\numberline {5.2.1}Dyck words}{29}{subsection.5.2.1}%
\contentsline {subsection}{\numberline {5.2.2}Mountain ranges}{29}{subsection.5.2.2}%
\contentsline {subsection}{\numberline {5.2.3}Lattice paths}{29}{subsection.5.2.3}%
\contentsline {section}{\numberline {5.3}Eulerian numbers}{29}{section.5.3}%
\contentsline {part}{II\hspace {1em}Advanced}{31}{part.2}%
\contentsline {chapter}{\numberline {6}Generating functions}{33}{chapter.6}%
\contentsline {section}{\numberline {6.1}Ordinary generating functions}{34}{section.6.1}%
\contentsline {section}{\numberline {6.2}Exponential generating functions}{35}{section.6.2}%
\contentsline {section}{\numberline {6.3}Lambert series}{35}{section.6.3}%
\contentsline {section}{\numberline {6.4}Bell series}{35}{section.6.4}%
\contentsline {section}{\numberline {6.5}Dirichlet series}{35}{section.6.5}%
\contentsline {chapter}{\numberline {7}Analytical combinatorics}{37}{chapter.7}%
\contentsline {section}{\numberline {7.1}Generating functions}{37}{section.7.1}%
\contentsline {section}{\numberline {7.2}Circle method}{37}{section.7.2}%
\contentsline {section}{\numberline {7.3}Saddle-point method}{37}{section.7.3}%
\contentsline {subsection}{\numberline {7.3.1}Dobiński's formula}{37}{subsection.7.3.1}%