• i. Counting Fundamentals: Counting rules, binomials (Pascal table, huge $n$), bounded stars and bars, permutation toolkit (inverse, cycles, rank/unrank) and the twelvefold way.
• ii. Binomial Identities: The identity table with proofs, Lucas/Kummer/parity, and row prefix sums (single, offline with Mo) and residue-class sums of binomials.
• iii. Inclusion-Exclusion and Derangements: Inclusion–exclusion with Bonferroni, surjections and Möbius forms, derangements, rook polynomials, ménage, Ryser permanent and Möbius inversion on posets.
• iv. Catalan Numbers and Lattice Paths: Catalan numbers and their bijections, reflection and the cycle lemma, ballot, k-th bracket sequence, paths with obstacles and between two lines, Fuss–Catalan, Narayana.
• v. Stirling, Bell and Partition Numbers: Stirling numbers, Bell, Lah, Fubini, Eulerian, Motzkin, Schröder, Delannoy, integer partitions (pentagonal, bounded, distinct) and q-binomials with q-Lucas.
• vi. Burnside, Pólya and Permutation Groups: Burnside and Pólya: necklaces, bracelets, grids, cubes, graphs up to isomorphism; Schreier–Sims, permutation roots and encodings.
• vii. Generating Functions: Encode a sequence $(a_0,a_1,a_2,\dots )$ as the formal power series
• viii. Counting Graphs and Trees: Labelled graphs, connected graphs, DAGs, even graphs, Cayley and forests, functional graphs, and unlabelled rooted and free trees (Otter).
• ix. Young Tableaux and RSK: Insertion and recording tableaux of a sequence; the shape gives, by Greene’s theorem, the maximum total length of k disjoint increasing subsequences.