Twelvefold way

The twelvefold way organizes counting problems that assign objects to boxes according to whether objects and boxes are distinguishable and whether empty boxes are allowed.

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Set partition: A division of a set into nonempty, disjoint subsets whose union is the original set. Partitions describe assignments to indistinguishable, nonempty boxes.
Exponential formula: A generating-function principle that assembles labeled structures from sets of labeled components. It converts partitions of distinguishable objects into exponential generating functions.
Integer partition: A representation of a positive integer as a sum of positive integers, with order ignored. Occupancy counts of indistinguishable boxes lead to integer partitions when the box count varies.
Gian-Carlo Rota: An Italian-American mathematician who shaped modern combinatorics and developed the umbral calculus. Rota presented the systematic organization of these twelve distribution problems.
Stirling number of the second kind: The number of ways to partition a set of n elements into k nonempty, unlabeled subsets. These numbers count distributions among indistinguishable boxes when none may be empty.
Ordinary generating function: A power series whose coefficients encode counts indexed by a nonnegative integer. Its factors track occupancies when objects or boxes are indistinguishable.
Rook polynomial: A polynomial whose coefficients count nonattacking placements of rooks on a board. Restricted box assignments can be counted by encoding forbidden object-box pairs on a board.
Combinatorial theory: The mathematical study of finite or discrete structures and the ways they can be counted. The twelvefold way is a compact taxonomy within enumerative combinatorics.
Bell number: The number of partitions of a finite set, summed over all possible numbers of blocks. Summing the fixed-box partition counts gives the total over every number of occupied boxes.
Exponential generating function: A power series with coefficients divided by factorials, used to count labeled structures. It naturally models assignments of distinguishable objects to labeled boxes.