Examples

This gallery walks through every public feature of mathematicskit.combinatorics: permutation/combination counting and generation, Pascal’s triangle, integer partitions, the inclusion-exclusion principle, and Stirling/ Catalan/Bell numbers.

See also the narrative tutorial:

• Counting, Primes, and Probability

Each script in this gallery is self-contained and can be run directly with python examples/combinatorics/<section>/<script>.py.

Sections

• counting – permutation/combination counting and sequence generation, and multinomial coefficients.

• pascals_triangle – the binomial-coefficient recurrence.

• partitions – the partition function, enumeration, and Young diagrams.

• inclusion_exclusion – the inclusion-exclusion principle and derangements.

• special_numbers – Stirling numbers, Catalan numbers, and Bell numbers.

Counting

Permutation/combination counting and generation, and multinomial coefficients.

Counting poker hands

Counting poker hands

Gersonides’ counting formulas

Gersonides' counting formulas

Combinatorial designs

Latin squares and orthogonal pairs.

Euler’s officers: orthogonal Latin squares

Euler's officers: orthogonal Latin squares

Extremal combinatorics

Ramsey’s theorem and the Erdős-Szekeres theorem: order that must appear in any large enough structure.

Ramsey’s theorem: R(3,3) = 6

Ramsey's theorem: R(3,3) = 6

The Erdős-Szekeres theorem: monotone subsequences

The Erdős-Szekeres theorem: monotone subsequences

Inclusion-exclusion

The inclusion-exclusion principle and derangements.

The hat-check problem: derangements via inclusion-exclusion

The hat-check problem: derangements via inclusion-exclusion

Matchings

Hall’s marriage theorem and bipartite matchings.

Hall’s marriage theorem

Hall's marriage theorem

Necklaces

Pólya enumeration: counting colorings up to symmetry.

Pólya’s enumeration theorem: necklaces and bracelets

Pólya's enumeration theorem: necklaces and bracelets

Integer partitions

The partition function, enumeration, and Young diagrams.

Integer partitions, the partition function, and Young diagrams

Integer partitions, the partition function, and Young diagrams

Pascal’s triangle

The binomial-coefficient recurrence, built by hand.

Pascal’s triangle vs. scipy-computed binomial coefficients

Pascal's triangle vs. scipy-computed binomial coefficients

Integer sequences

Fibonacci numbers, Bernoulli numbers and sums of powers, and Gray codes.

Fibonacci’s rabbits and domino tilings

Fibonacci's rabbits and domino tilings

Jacob Bernoulli’s numbers and sums of powers

Jacob Bernoulli's numbers and sums of powers

Gray codes: counting one bit at a time

Gray codes: counting one bit at a time

Special numbers

Stirling numbers, Bell numbers, and Catalan numbers.

Stirling’s numbers: cycles, set partitions, and Bell numbers

Stirling's numbers: cycles, set partitions, and Bell numbers

Catalan numbers: Euler’s polygon triangulations and Catalan’s brackets

Catalan numbers: Euler's polygon triangulations and Catalan's brackets

Labeled trees

Cayley’s formula and Prüfer codes.

Cayley’s formula via Prüfer codes

Cayley's formula via Prüfer codes