By Richard P. Stanley

This e-book, the 1st of a two-volume easy advent to enumerative combinatorics, concentrates at the conception and alertness of producing services, a basic instrument in enumerative combinatorics. Richard Stanley covers these components of enumerative combinatorics with the best functions to different parts of arithmetic. The 4 chapters are dedicated to an available creation to enumeration, sieve methods--including the primary of Inclusion-Exclusion, partly ordered units, and rational producing features. quite a few routines, just about all with options, increase the textual content and supply access into many components now not coated without delay. Graduate scholars and learn mathematicians who desire to observe combinatorics to their paintings will locate this an authoritative reference.

Show description

Read or Download Enumerative combinatorics, vol. 1 PDF

Similar combinatorics books

Proofs from THE BOOK

This revised and enlarged 5th version positive aspects 4 new chapters, which comprise hugely unique and pleasant proofs for classics akin to the spectral theorem from linear algebra, a few newer jewels just like the non-existence of the Borromean jewelry and different surprises. From the Reviews". .. within PFTB (Proofs from The publication) is certainly a glimpse of mathematical heaven, the place shrewdpermanent insights and gorgeous rules mix in marvelous and excellent methods.

Combinatorial Algebraic Geometry: Levico Terme, Italy 2013, Editors: Sandra Di Rocco, Bernd Sturmfels

Combinatorics and Algebraic Geometry have loved a fruitful interaction because the 19th century. Classical interactions contain invariant conception, theta capabilities and enumerative geometry. the purpose of this quantity is to introduce fresh advancements in combinatorial algebraic geometry and to technique algebraic geometry with a view in the direction of functions, equivalent to tensor calculus and algebraic facts.

Finite Geometry and Combinatorial Applications

The projective and polar geometries that come up from a vector house over a finite box are relatively valuable within the development of combinatorial gadgets, similar to latin squares, designs, codes and graphs. This e-book offers an advent to those geometries and their many functions to different parts of combinatorics.

Extra info for Enumerative combinatorics, vol. 1

Sample text

Define a map θ : 2S → {0, 1}n by θ(T ) = (ε1 , ε2 , . . , εn ), where 1, if xi ∈ T εi = 0, if xi ∈ T . For example, if n = 5 and T = {x2 , x4 , x5 }, then θ (T ) = (0, 1, 0, 1, 1). Most readers will realize that θ (T ) is just the characteristic vector of T . It is easily seen that θ is a bijection, so that we have given a combinatorial proof that #2S = 2n . Of course, there are many alternative proofs of this simple result, and many of these proofs could be regarded as combinatorial. Now define Sk (sometimes denoted S (k) or otherwise, and read “S choose k”) to be the set of all k-element subsets (or k-subsets) of S, and define nk = # Sk , read “n choose k” (ignore our previous use of the symbol nk ) and called a binomial coefficient.

Hence, N (n, k) = n+k−1 k−1 . A further variant is the enumeration of N-solutions (that is, solutions where each variable lies in N) to x1 + x2 + · · · + xk ≤ n. Again we use a standard technique, namely, introducing a slack variable y to convert the inequality x1 + x2 + · · · + xk ≤ n to the equality x1 + x2 + · · · + xk + y = n. An N-solution to this equation is a weak (k + 1)-composition of n, so the number N (n, k + 1) of such solutions is n+(k+1)−1 = n+k k k . A k-subset T of an n-set S is sometimes called a k-combination of S without repetitions.

Moreover, we denote the polynomial 1 + q + · · · + q n−1 = (1 − q n )/(1 − q) by (n) and call it “the q-analogue of n,” so that (n)! = (1)(2) · · · (n). In general, a q-analogue of a mathematical object is an object depending on the variable q that “reduces to” (an admittedly vague term) the original object when we set q = 1. 4 Descents 31 be interpreted in terms of subspaces of finite-dimensional vector spaces over the finite field Fq . For instance, n! is the number of sequences ∅ = S0 ⊂ S1 ⊂ · · · ⊂ Sn = [n] of subsets of [n].

Download PDF sample

Rated 4.72 of 5 – based on 49 votes