Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements

Details Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements

2006-09-01

arxiv.org/abs/math/0609051v1

J. Combinatorial Theory Ser. A, 114 (2007), no. 1, 97-109.

Mathematics

Combinatorics

13 pp

Scientific paper

Hyperplanes of the form x_j = x_i + c are called affinographic. For an affinographic hyperplane arrangement in R^n, such as the Shi arrangement, we study the function f(M) that counts integral points in [1,M]^n that do not lie in any hyperplane of the arrangement. We show that f(M) is a piecewise polynomial function of positive integers M, composed of terms that appear gradually as M increases. Our approach is to convert the problem to one of counting integral proper colorations of a rooted integral gain graph. An application is to interval coloring in which the interval of available colors for vertex v_i has the form [(h_i)+1,M]. A related problem takes colors modulo M; the number of proper modular colorations is a different piecewise polynomial that for large M becomes the characteristic polynomial of the arrangement (by which means Athanasiadis previously obtained that polynomial). We also study this function for all positive moduli.

Affiliated with

Also associated with

No associations

Rating

Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-109781