For instance, there are 4 possible ways to write the vector (3,1,-4) in terms of e12,e13,e23 (this is the root system A2).

(3,1,-4) = 0 * e12 + 3 * e13 + 1 * e23

= 1 * e12 + 2 * e13 + 2 * e23

= 2 * e12 + 1 * e13 + 3 * e23

= 3 * e12 + 0 * e13 + 4 * e23

It is known that Konstant's partition function is a piecewise polynomial of degree {n-1 \choose 2}. The number of pieces equals 1 for n=1, 2 for n=2, 7 for n=3, 48 for n = 4, 820 for n = 5, and 44288 for n = 6. It is not known how this sequence continues. This website provides a way to evaluate those polynomials pieces up to n = 5. For details on how the polynomials were generated please consult the paper Algebraic Unimodular Counting by Jesus De Loera and Bernd Sturmfels. If you are interested to look at a particular polynomial please send us email. We have a MAPLE subroutine of the 48 polynomials for A4 available here.

You are also invited to explore the closely related enumeration problem of COUNTING CONTINGENCY TABLES.

by the positive roots of A_n (for example

Oct/05/2000