Glossary

The total count of ways to partition a set into any number of unlabeled, nonempty blocks.

A pairing between two sets that matches every element of one to exactly one element of the other, with nothing left over.

Proving two counts equal by building an explicit reversible pairing between the two sets.

A graph whose vertices split into two sides with every edge crossing between them.

The count of diagonal-respecting lattice paths, also counting balanced parentheses and binary tree shapes.

The count of labeled trees on n vertices, exactly n to the power n minus 2.

The polynomial equation whose roots determine a linear recurrence's closed-form solution.

The fewest colors needed to color a graph so no edge joins two same-colored vertices.

A polynomial in k counting exactly how many proper colorings a graph has with k colors available.

An arrangement around a circle where a rotation of the same order counts as identical.

Reading a_n back off a generating function as the coefficient of x^n.

An ordered sum of positive integers equaling n, where the order of the parts matters.

The partition formed by reading a Ferrers diagram's columns instead of its rows.

The list of every vertex's degree, sorted, used to check whether a degree list can come from a real graph.

One of four counting cases decided by whether the objects and the boxes are each identical or distinct.

Counting one set of objects two different, valid ways, proving the two resulting expressions equal.

A row of dots per part, longest row on top, that makes a partition's structure visible.

A formal power series that files a whole sequence as coefficients of x^n.

A set of edges with no shared endpoint, pairing up vertices without overlap.

A bipartite graph has a matching saturating one side exactly when every subset's neighborhood is large enough.

A function where no two inputs share an output, counted by a falling factorial.

An unordered sum of positive integers equaling n, unlike a composition where order matters.

A graph on distinguishable, numbered vertices, so relabeling produces a different graph.

A path on a grid moving only right or up, counted by choosing which steps are which.

A three-case rule that reads off a divide-and-conquer recurrence's growth rate directly from its parameters.

Identifying what is being chosen or arranged, from what pool, before touching a formula.

An arrangement of items where some are identical copies, counted by dividing out each repeated group's internal reorderings.

A circular arrangement where both rotation and reflection are considered the same, dividing the circular count by an extra 2.

Counting a bigger, easier set where every real outcome is hit the same number of times, then dividing that fixed factor away.

The count of all partitions of n, with no known simple closed formula.

The finite category every object in a pigeonhole argument is sorted into, rarely handed over explicitly by the problem.

A diagram of a recurrence's subproblems, level by level, used to total the cost by hand.

A root the characteristic equation has more than once, needing an extra factor of n in the solution.

How many distinct spanning trees a given graph contains, computed by the matrix-tree theorem.

Distributing identical items into labeled bins, modeled as a row of stars split by dividing bars.

The count of ways to split n labeled elements into exactly k unlabeled, nonempty blocks.

A function that hits every target at least once, counted by inclusion-exclusion.