# tMacaulayExpansion -- compute the t-Macaulay expansion of a positive integer

## Synopsis

• Usage:
tMacaulayExpansion(a,n,d,t)
• Inputs:
• a, a positive integer to be expanded
• n, a positive integer that identifies the number of indeterminates
• d, a positive integer that identifies the degree
• t, a positive integer that idenfies the t-spread contest
• Optional inputs:
• Shift => ..., default value false, optional boolean argument for tMacaulayExpansion
• Outputs:
• a list, pairs of positive integers representing the d-th (shifted) t-Macaulay expansion of a with a given n

## Description

Given four positive integers (a,n,d,t) there is a unique expression of a as a sum of binomials $a=\binom{a_d}{d} + \binom{a_{d-1}}{d-1} + \cdots + \binom{a_j}{j}.$ where $a_i > a_{i-1} > \cdots > a_j > j >= 1.$
If the optional parameter Shift is true, then the method tMacaulayExpansion(a,n,d,t,Shift=>true) returns the shifted t-Macaulay expansion of a, that is, $a^{(d)}=\binom{a_d}{d+1} + \binom{a_{d-1}}{d} + \cdots + \binom{a_j}{j+1}.$To obtain the sum of the binomial coefficients represented in the output list, one can use the method solveBinomialExpansion.

Examples:

 i1 : tMacaulayExpansion(50,12,2,1) o1 = {{10, 2}, {5, 1}} o1 : List i2 : tMacaulayExpansion(50,12,2,1,Shift=>true) o2 = {{10, 3}, {5, 2}} o2 : List i3 : tMacaulayExpansion(50,12,2,2,Shift=>true) o3 = {{9, 3}, {4, 2}} o3 : List