Enumerative Combinatorics, Volume 1¶
Stanley, R. P. (2011). Enumerative Combinatorics, Volume 1. Cambridge University Press.
Cited by¶
8 citations across 8 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Bijectivity
- Take the claim that the number of subsets of an \(n\)-element set equals \(2^n\).
This sourceStandard treatment of bijective proof, including the membership-to-binary-string bijection establishing that an n-element set has 2^n subsets, and the principle that an explicit bijection exhibits a counting identity.
- Take the claim that the number of subsets of an \(n\)-element set equals \(2^n\).
- Order
- The type lattice of a programming language is a partial order under the subtype relation that is often a lattice with
⊥(the empty type, "no values") and⊤(the universal type, "any value"), with combinatorial-order machinery developed by Stanley (2011)This sourceComprehensive modern treatment of combinatorial order theory; canonical source for poset enumeration, generating functions on lattices, and combinatorial structure of finite ordered sets.
- The type lattice of a programming language is a partial order under the subtype relation that is often a lattice with
- Union
- The overlap-collapsing behavior is the heart of the example and the source of its most-cited formula: an element in several \(A_i\) is counted once in the union, so the cardinality is not \(\sum_i |A_i|\) (which double-counts the overlaps) but the inclusion–exclusion alternating sum \(\left|\bigcup_i A_i\right| = \sum |A_i| - \sum |A_i \cap A_j| + \sum |A_i \cap A_j \cap A_k| - \cdots\).
This sourceStates and proves the inclusion–exclusion principle, computing the cardinality of a union of overlapping sets as the alternating sum over their intersections.
- The overlap-collapsing behavior is the heart of the example and the source of its most-cited formula: an element in several \(A_i\) is counted once in the union, so the cardinality is not \(\sum_i |A_i|\) (which double-counts the overlaps) but the inclusion–exclusion alternating sum \(\left|\bigcup_i A_i\right| = \sum |A_i| - \sum |A_i \cap A_j| + \sum |A_i \cap A_j \cap A_k| - \cdots\).
Domain-specific¶
- Constant Term
- Order polynomial
- Pascal's rule
- q-Analog
- Mathematical practice reserves the name for deformations that organize real structure: coefficients record a statistic, finite-field objects replace set-theoretic objects, a basic hypergeometric identity extends a classical identity, a q-difference operator replaces a derivative, or evaluation at roots of unity exposes symmetry
This sourceStanley's Enumerative Combinatorics I supports the statistic criterion (Corollary 1.3.13 and section 1.10, 'Two q-analogues of permutations'); the other criteria in this sentence are carried by other works. Stanley, Enumerative Combinatorics I, Corollary 1.3.13: the sum of q^inv(w) over the permutations of n is the q-factorial.
Supported in partVerified against the source
- Mathematical practice reserves the name for deformations that organize real structure: coefficients record a statistic, finite-field objects replace set-theoretic objects, a basic hypergeometric identity extends a classical identity, a q-difference operator replaces a derivative, or evaluation at roots of unity exposes symmetry
- Twelvefold Way
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