# allHilbertSequences -- compute all Hilbert sequences of quotients in an exterior algebra

## Synopsis

• Usage:
allHilbertSequences E
• Inputs:
• E, an exterior algebra
• Outputs:
• a list, all Hilbert sequences of quotients of E

## Description

A sequence is called a Hilbert sequence whether it satisfies the Kruskal-Katona theorem in the exterior algebra E.

Example:

 i1 : E=QQ[e_1..e_4,SkewCommutative=>true] o1 = E o1 : PolynomialRing, 4 skew commutative variables i2 : allHilbertSequences E o2 = {{1, 4, 6, 4, 1}, {1, 4, 6, 4, 0}, {1, 4, 6, 3, 0}, {1, 4, 6, 2, 0}, {1, ------------------------------------------------------------------------ 4, 6, 1, 0}, {1, 4, 6, 0, 0}, {1, 4, 5, 2, 0}, {1, 4, 5, 1, 0}, {1, 4, ------------------------------------------------------------------------ 5, 0, 0}, {1, 4, 4, 1, 0}, {1, 4, 4, 0, 0}, {1, 4, 3, 1, 0}, {1, 4, 3, ------------------------------------------------------------------------ 0, 0}, {1, 4, 2, 0, 0}, {1, 4, 1, 0, 0}, {1, 4, 0, 0, 0}, {1, 3, 3, 1, ------------------------------------------------------------------------ 0}, {1, 3, 3, 0, 0}, {1, 3, 2, 0, 0}, {1, 3, 1, 0, 0}, {1, 3, 0, 0, 0}, ------------------------------------------------------------------------ {1, 2, 1, 0, 0}, {1, 2, 0, 0, 0}, {1, 1, 0, 0, 0}, {1, 0, 0, 0, 0}, {0, ------------------------------------------------------------------------ 0, 0, 0, 0}, {-1, 0, 0, 0, 0}} o2 : List

• lexIdeal -- compute the lex ideal with a given Hilbert function in an exterior algebra
• isHilbertSequence -- whether the given sequence is a Hilbert sequence

## Ways to use allHilbertSequences :

• "allHilbertSequences(Ring)"

## For the programmer

The object allHilbertSequences is .