Enumerative Combinatorics

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10 domain-specific abstractions whose origin domain is Enumerative Combinatorics.

• Catalan number — The integer sequence C_n=(1/(n+1)) binomial(2n,n) that counts many recursively nested structures such as balanced parentheses, polygon triangulations and binary trees.
• Combinatorial number system — A bijective representation of nonnegative integers as fixed-size combinations through sums of binomial coefficients, enabling direct ranking and unranking without listing earlier combinations.
• Derangement — A permutation with no fixed points, so every element moves away from its original position.
• Eulerian number — A permutation-counting number classified by an exact number of ascents or, under shifted conventions, descents.
• Hyperharmonic number — A recursively iterated family of partial sums beginning with reciprocals and extending the ordinary harmonic numbers by an order parameter.
• Lobb number — A two-parameter ballot number counting prefixes of balanced-parenthesis paths with a specified excess of open parentheses, generalizing Catalan numbers.
• Motzkin number — A sequence whose nth term counts noncrossing chord matchings on n labeled points in convex position, allowing unmatched points.
• Schröder number — A sequence counting diagonal-bounded lattice paths with horizontal, vertical and diagonal steps, equivalently several recursively decomposable combinatorial structures.
• Stirling numbers of the second kind — The numbers S(n,k) counting partitions of an n-element labeled set into exactly k nonempty unlabeled blocks.
• Stirling transform — The invertible sequence transform that weights source terms by Stirling numbers of the second kind, with inverse coefficients given by signed first-kind Stirling numbers.