macaulayExpansion -- compute the Macaulay expansion of a positive integer

Synopsis

Usage:

macaulayExpansion(a,i)
Inputs:
- a, a positive integer
- i, a positive integer
Optional inputs:
- Shift => ..., default value false, optional argument for macaulayExpansion
Outputs:
- a list, pairs of positive integers representing the i-th (shifted) Macaulay expansion of a

Description

Given a pair of positive integers (a,i) there is a unique expression of a as a sum of binomials a=binomial(a_i,i) + binomial(a_{i-1},i-1) + ... + binomial(a_j,j) where a_i > a_{i-1} > ... > a_j > j >= 1.

Examples:

i1 : macaulayExpansion(8,4)

o1 = {{5, 4}, {3, 3}, {2, 2}, {1, 1}}

o1 : List

i2 : macaulayExpansion(3,1,Shift=>false)

o2 = {{3, 1}}

o2 : List

i3 : macaulayExpansion(8,4,Shift=>true)

o3 = {{5, 5}, {3, 4}, {2, 3}, {1, 2}}

o3 : List

i4 : macaulayExpansion(3,1,Shift=>true)

o4 = {{3, 2}}

o4 : List

Ways to use macaulayExpansion :

macaulayExpansion(ZZ,ZZ)

For the programmer

The object macaulayExpansion is a method function with options.

macaulayExpansion -- compute the Macaulay expansion of a positive integer

Synopsis

Description

See also

Ways to use macaulayExpansion :

For the programmer