mathjs/lib/function/combinatorics/bellNumbers.js

'use strict';

function factory (type, config, load, typed) {
  var add = load(require('../arithmetic/add'));
  var stirlingS2 = load(require('./stirlingS2'));
  var isNegative = load(require('../utils/isNegative'));
  var isInteger = load(require('../utils/isInteger'));

  /**
   * The Bell Numbers count the number of partitions of a set. A partition is a pairwise disjoint subset of S whose union is S.
   * bellNumbers only takes integer arguments.
   * The following condition must be enforced: n >= 0
   *
   * Syntax:
   *
   *   math.bellNumbers(n)
   *
   * Examples:
   *
   *    math.bellNumbers(3); // returns 5;
   *    math.bellNumbers(8); // returns 4140;
   *
   * See also:
   *
   *    stirlingS2
   *
   * @param {Number | BigNumber} n    Total number of objects in the set
   * @return {Number | BigNumber}     B(n)
   */
  var bellNumbers = typed('bellNumbers', {
    'number | BigNumber': function (n) {

      if (!isInteger(n) || isNegative(n)) {
        throw new TypeError('Non-negative integer value expected in function bellNumbers');
      }

      // Sum (k=0, n) S(n,k).
      var result = 0;
      for(var i = 0; i <= n; i++) {
        result = add(result, stirlingS2(n, i));
      }

      return result;
    }
  });

  bellNumbers.toTex = '\\mathrm{B}_{${args[0]}}';

  return bellNumbers;
}

exports.name = 'bellNumbers';
exports.factory = factory;