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A positive formula for the Ehrhart-like polynomials from root system chip-firing. (arXiv:1803.08472v1 [math.CO])
来源于:arXiv
In earlier work in collaboration with Pavel Galashin and Thomas McConville we
introduced a version of chip-firing for root systems. Our investigation of root
system chip-firing led us to define certain polynomials analogous to Ehrhart
polynomials of lattice polytopes, which we termed the symmetric and truncated
Ehrhart-like polynomials. We conjectured that these polynomials have
nonnegative integer coefficients. Here we affirm "half" of this positivity
conjecture by providing a positive, combinatorial formula for the coefficients
of the symmetric Ehrhart-like polynomials. This formula depends on a subtle
integrality property of slices of permutohedra which may be of independent
interest. We also discuss how our formula very naturally suggests a conjecture
for the coefficients of the truncated Ehrhart-like polynomials that turns out
to be false in general, but which may hold in some cases. 查看全文>>