Special Numbers

Abstract

Iverson notation Iverson (the deviser of APL) invented the following handy bracket notation: [statement] = 1 if statement is true; 0 if statement is false. Thus, for example, [i = j] = 1 if i = j and [i = j] = 0 if i = j: [i = j] = δ ij , the Kronecker delta. Binomial Coefficients Here is the general definition of the binomial coefficient r k : r k := r k k! for real r and integer k ≥ 0; it is defined to be zero for real r and integer k < 0. For integer k ≥ 0, r k = [x k ](1 + x) r := coefficient of x k in the expansion of (1 + x) r. For integers k and n with 0 ≤ k ≤ n, n k , read " n choose k, " counts the number of ways to select a subset of k objects from a set of n objects. Stirling Numbers of the Second Kind For integers k and n ≥ 0, n k = coefficient of x k in x n written as a factorial polynomial. Thus, n k = 0 for integer k < 0 as well as for integer k > n, and for integer n ≥ 0, x n = n k=0 n k x k = k∈Z n k x k. Stirling numbers of the second kind satisfy the recurrence relation n k = k n − 1 k + n − 1 k − 1 for integer n > 0 and integer k with boundary conditions n 0 = [n = 0] and n n = 1. The number n k , read " n subset k, " counts the number of partitions of a set of n elements into k nonempty subsets. Other notations are used for Stirling numbers of the second kind. The classic handbook AMS 55 [1] uses a notation like S (k) n but with a fancy calligraphic " S. " Spiegel [3, p. 7] uses the notation S n k , while Combinatorics [4, p. 37] uses the notation S(n, k).