# Kostant's partition function for the root system A_n

INTRODUCTION: The Kostant partition function for the root system A_n takes an integer vector of length n+1 whose coordinates sum to zero and returns the numbers of ways of writing it as nonnegative integer linear combination of the {n+1 \choose 2 } vectors e12 = (1,-1,0,...,0), e13 = (1,0,-1,...,0), ..., e_{n,n+1} = (0,...,0,1,-1). Equivalently, we are counting integral flows on a complete directed graph K_(n+1). Here you can evaluate Kostant's partition function for n<=5

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.

Input a vector of length n+1 for n=3,4,5 that lies in the linear space generated
by the positive roots of A_n (for example 3, 3,-2, 1, -5 if working with A4)
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Oct/05/2000

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