multinomial -- a formula for the multinomial coefficient

Synopsis

Usage:

multinomial(tal)

multinomial(l)

multinomial(p)
Inputs:
- p, an instance of the type Partition, a partition
- l, a list, a list of non negative numbers
- tal, a tally, a tally from a list
Outputs:
- an integer, the multinomial coefficient of the given list

Description

The multinomial coefficient is a generalization of the binomial coefficient. Given a list of number $k_1,\ldots,k_l$, the multinomial coefficient is $n!/(k_1!\ldots,k_l!)$ where $n = \sum k_i$. The multinomial coefficient is calculated because it gives the numbers of tabloids for a given partition.

The list of numbers used to calculate the multinomial can be given as a list, a partition or a tally. This last option was added to optimize this calculation.

i1 : p = new Partition from {2,2}

o1 = Partition{2, 2}

o1 : Partition

i2 : tabloids p

o2 = {| 0 1 |, | 0 2 |, | 0 3 |, | 1 2 |, | 1 3 |, | 2 3 |}
      | 2 3 |  | 1 3 |  | 1 2 |  | 0 3 |  | 0 2 |  | 0 1 |

o2 : TableauList

i3 : multinomial {2,2}

o3 = 6

i4 : multinomial tally {2,2}

o4 = 6

Ways to use multinomial:

multinomial(List)
multinomial(Partition)
multinomial(Tally)

For the programmer

The object multinomial is a method function.