Ramsey theory

Sufficiently large structures force specified regular substructures under suitable coloring conditions.

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Burnside's lemma
For a finite group action, orbit count equals the average number of fixed points.
Integer partitions
A partition writes a positive integer as an unordered sum of positive integers.
Binomial theorem
Binomial coefficients give the expansion of an integer nonnegative power of a sum.
Catalan numbers
Catalan numbers count several recursively structured objects, including balanced parenthesis strings.