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# expo -- returns a set corresponding to the basis of a divided power

## Synopsis

• Usage:
B = expo(c,N)
B = expo(c,L)
• Inputs:
• Outputs:
• B, a list, partitions with c non-negative parts

## Description

The form expo(c,N) returns partitions of N with c non-negative parts. The form expo(c, L) returns partitions with non-negative parts that are componentwise <= L (and any sum <= sum L).

The list expo(c,N) may be thought of as the list of exponent vectors of the monomials of degree N in c variables. This is used in the construction of the Eisenbud-Shamash resolution.

The list expo(c, L), on the other hand, may be thought of as the list of divisors of e^L = e_0^{L_0} ... e_c^{L_c}. This is used in the construction of the higher homotopies on a complex.

 i1 : expo(3,5) o1 = {{5, 0, 0}, {4, 1, 0}, {4, 0, 1}, {3, 2, 0}, {3, 1, 1}, {3, 0, 2}, {2, ------------------------------------------------------------------------ 3, 0}, {2, 2, 1}, {2, 1, 2}, {2, 0, 3}, {1, 4, 0}, {1, 3, 1}, {1, 2, 2}, ------------------------------------------------------------------------ {1, 1, 3}, {1, 0, 4}, {0, 5, 0}, {0, 4, 1}, {0, 3, 2}, {0, 2, 3}, {0, 1, ------------------------------------------------------------------------ 4}, {0, 0, 5}} o1 : List i2 : expo(3, {3,2,1}) o2 = {{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {2, 0, 0}, {1, 1, 0}, {1, ------------------------------------------------------------------------ 0, 1}, {0, 2, 0}, {0, 1, 1}, {3, 0, 0}, {2, 1, 0}, {2, 0, 1}, {1, 2, 0}, ------------------------------------------------------------------------ {1, 1, 1}, {0, 2, 1}, {3, 1, 0}, {3, 0, 1}, {2, 2, 0}, {2, 1, 1}, {1, 2, ------------------------------------------------------------------------ 1}, {3, 2, 0}, {3, 1, 1}, {2, 2, 1}} o2 : List