Mathematics / core
Enumerative combinatorics
Translate finite structures into recurrences, generating functions, inversion formulas, and bijective arguments.
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Inside this module
Module 01 core
Enumerative combinatorics
Translate finite structures into recurrences, generating functions, inversion formulas, and bijective arguments.
- 01
Combinatorial classes and bijections
Specify finite structures precisely and prove equal counts by constructing reversible encodings.
27 min - 02
Recurrence relations and counting
Derive recurrences from reversible structural cases and supply the initial data that make them determine a sequence.
28 min - 03
Ordinary generating functions
Encode a sequence as a formal power series and translate additive size decompositions into coefficient identities.
30 min - 04
Exponential generating functions
Account for distinct labels with factorial-normalized coefficients and the exponential formula for sets of components.
31 min - 05
Partitions and Ferrers diagrams
Count unordered integer decompositions using multiplicities, conjugate diagrams, and coefficientwise finite products.
29 min - 06
Stirling numbers and set partitions
Count partitions of distinct labels by block number, using a structural recurrence and an exponential generating function.
30 min - 07
Catalan objects and lattice paths
Derive Catalan counts by a first-return decomposition, a reflection bijection, and reversible path encodings.
32 min - 08
Burnside's lemma and orbit counting
Count structures up to a specified finite symmetry group by averaging fixed structures rather than assuming equal orbit sizes.
31 min - 09
Möbius inversion on posets
Recover local contributions from cumulative counts using the recursively defined inverse of interval summation.
33 min - 10
Lagrange inversion and tree enumeration
Extract coefficients from implicit formal equations and connect the formula to rooted plane and labeled trees.
35 min