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.
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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].