Enumerative Combinatorics, Volume 1

Stanley, R. P. (2011). Enumerative Combinatorics, Volume 1. Cambridge University Press.

Type

Book

Intellectual base

Review or monograph

Year

2011

DOI

10.1017/cbo9781139058520

ISBN

9781107015425

Link

https://doi.org/10.1017/CBO9781139058520

Cited from

mathematics

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.

• 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.

• Union
  • … 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.

Domain-specific

• Constant Term
  • It is a linear coefficient-extraction operator and corresponds to the multiplicative identity monomial `1`.
• Order polynomial
  • … between constitutive identity and a convenient observable transfers from A canonical example satisfies the full defining rule of Order polynomial with assumptions and conventions explicit. to A careful use of Order polynomial tests the constitutive rule and nearest confusable rather than relying on the label alone..
• Pascal's rule
  • It says that the number of \(k\)-element subsets of an \(n\)-element set is the sum of those omitting one distinguished element and those containing it.
• 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.

• Twelvefold Way
  • It crosses two binary symmetry choices—whether objects are distinguishable and whether boxes are distinguishable—with three mapping constraints—unrestricted, injective, or surjective—to form twelve standard counting problems.

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• https://www.cambridge.org/core/books/enumerative-combinatorics/0FF8E3D08C814D5F46C0AC54F2D78441 ×1

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