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**Enumerative combinatorics. Volume 2.**
*(English)*
Zbl 0928.05001

Cambridge Studies in Advanced Mathematics. 62. Cambridge: Cambridge University Press. xii, 581 p. (1999).

[Compare Zbl 0608.05001 and Zbl 0889.05001 for the first edition of Volume 1]

This long awaited Volume 2 of Enumerative Combinatorics by Richard Stanley was well worth the wait! At 581 pages it is almost twice the length of Volume 1. Much of the increase in size has to do with the inclusion of even more exercises and solutions than in Volume 1. These carefully selected and organized problems along with their solutions is one of the assets of this book as a graduate text.

Volume 2 has just three chapters. Chapters 5 (Trees and the Composition of Generating Functions) and 6 (Algebraic, D-Finite, and Noncommutative Generating Functions) continue the development of generating functions initiated in the last chapter of Volume 1. Chapter 7 (Symmetric Functions) includes many traditional topics (e.g. Pólya Enumeration) but in this newly designed framework of symmetric functions. Seeing so many seemingly disjoint topics arranged and unified by one central theme makes reading Chapter 7 a joy.

This magnificent two-volume work is best described by a quote from Gian-Carlo Rota’s Forward to Volume 2: I find it impossible to predict when Richard Stanley’s two-volume exposition of combinatorics may be superseded. No one will dare try, let alone be able, to match the thoroughness of coverage, the care for detail, the definitiveness of proof, the elegance of presentation.

This long awaited Volume 2 of Enumerative Combinatorics by Richard Stanley was well worth the wait! At 581 pages it is almost twice the length of Volume 1. Much of the increase in size has to do with the inclusion of even more exercises and solutions than in Volume 1. These carefully selected and organized problems along with their solutions is one of the assets of this book as a graduate text.

Volume 2 has just three chapters. Chapters 5 (Trees and the Composition of Generating Functions) and 6 (Algebraic, D-Finite, and Noncommutative Generating Functions) continue the development of generating functions initiated in the last chapter of Volume 1. Chapter 7 (Symmetric Functions) includes many traditional topics (e.g. Pólya Enumeration) but in this newly designed framework of symmetric functions. Seeing so many seemingly disjoint topics arranged and unified by one central theme makes reading Chapter 7 a joy.

This magnificent two-volume work is best described by a quote from Gian-Carlo Rota’s Forward to Volume 2: I find it impossible to predict when Richard Stanley’s two-volume exposition of combinatorics may be superseded. No one will dare try, let alone be able, to match the thoroughness of coverage, the care for detail, the definitiveness of proof, the elegance of presentation.

Reviewer: J.E.Graver (Syracuse)

### MSC:

05-02 | Research exposition (monographs, survey articles) pertaining to combinatorics |

05A15 | Exact enumeration problems, generating functions |

05A16 | Asymptotic enumeration |

06A07 | Combinatorics of partially ordered sets |

05E05 | Symmetric functions and generalizations |

### Keywords:

enumerative combinatorics; generating functions; Pólya enumeration; exercises; symmetric functions
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\textit{R. P. Stanley}, Enumerative combinatorics. Volume 2. Cambridge: Cambridge University Press (1999; Zbl 0928.05001)

### Digital Library of Mathematical Functions:

Example 3 ‣ §26.18 Counting Techniques ‣ Properties ‣ Chapter 26 Combinatorial Analysis§26.5(i) Definitions ‣ §26.5 Lattice Paths: Catalan Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis

§26.6(ii) Generating Functions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis

§26.6(i) Definitions ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis

§26.6(iv) Identities ‣ §26.6 Other Lattice Path Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis

Chapter 26 Combinatorial Analysis

### Online Encyclopedia of Integer Sequences:

Number of irreducible representations of symmetric group S_n for which every matrix has determinant 1.a(n) = (6n)!n!/((3n)!(2n)!^2).

Triangle read by rows: T(n,m) = number of planar partitions of n with trace m.

a(n) = ((5n)!/(n!(2n)!))(gamma(1+n/2)/gamma(1+5n/2)).

a(n) = ((3*n)!/n!^2)*(Gamma(1+n/2)/Gamma(1+3n/2)).

Triangle read by rows: T(n,k) is the number of permutations p of [n] such that the length of the longest 2-stack sortable initial segment of p is equal to k.

Triangle read by rows: T(n,k) is number of Dyck n-paths with k UUDDs, 0 <= k <= n/2.

Number of Dyck paths of semilength n for which the number of ascents of length 1 is equal to the number of descents of length 1.

Central terms of the triangle in A119258.

Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and a Catalan number.

Number of ways to express n as the sum of an odd prime, a Lucas number and a Catalan number.

Characteristic array for partitions which define multiset repetition classes.

Triangle read by rows: Number of crossing set partitions of {1,2,...,n} into k blocks.

Integral factorial ratio sequence: a(n) = (8*n)!*n!/((4*n)!*(3*n)!*(2*n)!).

Partition array for the number of representative necklaces (only cyclic symmetry) with n beads, each available in n colors. Only the color type (signature) matters.

a(n) = Sum_{k = 0..n} (-1)^k * binomial(n,k) * binomial(2*n,k).

a(n) = 2^(2*n-1)*( binomial(3*n/2,n) + binomial((3*n-1)/2,n) ).

a(n) = (1/n!) * (5*n)!/(5*n/2)! * (3*n/2)!/(3*n)!.

a(n) = (1/n!) * (7*n)!/(7*n/2)! * (5*n/2)!/(5*n)!.

a(n) = (7*n)!*(3/2*n)!/((7*n/2)!*(3*n)!*(2*n)!).

a(n) = (9*n)!*(5/2*n)!/((9*n/2)!*(5*n)!*(2*n)!).

Number of normal semistandard Young tableaux whose shape is the integer partition with Heinz number n.

Number of rim-hook (or border-strip) tableaux whose shape is the integer partition with Heinz number n.

Number of rim-hook (or border-strip) tableaux of size n.

Heinz numbers of integer partitions with Durfee square of length > 2.

Difference between the length of the minimal square containing and the maximal square contained in the Young diagram of the integer partition with Heinz number n.

Number of integer partitions of n such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is 2.

Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is k.

Number of nonnegative integer solutions to 2*n = x_1 + x_2 + ... + x_n + 2*y_1 + 2*y_2 + ... + 2*y_n.

Number of nonnegative integer solutions to n = x_1 + x_2 + ... + x_(2*n) + 2*y_1 + 2*y_2 + ... + 2*y_(2*n).

a(n) = [x^n] (1 + x + x^2)^(3*n)/(1 + x)^(2*n).

a(n) = [x^n] (1 + x + x^2 + x^3)^(4*n)/(1 + x + x^2)^(3*n).

a(n) = [x^n] ( 1/((1 - x)^2*(1 - x^2)) )^n for n >= 1.

Square array read by ascending antidiagonals: T(n,k) = [x^(n*k)] ((1 + x)/(1 - x))^k for n, k >= 1.

Square array read by ascending antidiagonals: T(n,k) = 1/n * [x^k] 1/((1 - x)*(1 - x^2))^(n*k) for n, k >= 1.