Abstract
AbstractFor $${\varvec{a}} \in {\mathbb {R}}_{\ge 0}^{n}$$
a
∈
R
≥
0
n
, the Tesler polytope $${\text {Tes}}_{n}({\varvec{a}})$$
Tes
n
(
a
)
is the set of upper triangular matrices with non-negative entries whose hook sum vector is $${\varvec{a}}$$
a
. Motivated by a conjecture of Morales, we study the questions of whether the coefficients of the Ehrhart polynomial of $${\text {Tes}}_n(1,1,\dots ,1)$$
Tes
n
(
1
,
1
,
⋯
,
1
)
are positive. We attack this problem by studying a certain function constructed by Berline–Vergne and its values on faces of a unimodularly equivalent copy of $${\text {Tes}}_n(1,1,\dots ,1)$$
Tes
n
(
1
,
1
,
⋯
,
1
)
. We develop a method of obtaining the dot products appeared in formulas for computing Berline–Vergne’s function directly from facet normal vectors. Using this method together with known formulas, we are able to show Berline–Vergne’s function has positive values on codimension 2 and 3 faces of the polytopes we consider. As a consequence, we prove that the third and fourth coefficients of the Ehrhart polynomial of $${\text {Tes}}_{n}(1,\dots ,1)$$
Tes
n
(
1
,
⋯
,
1
)
are positive. Using the Reduction Theorem by Castillo and the second author, we generalize the above result to all deformations of $${\text {Tes}}_{n}(1,\dots ,1)$$
Tes
n
(
1
,
⋯
,
1
)
including all the integral Tesler polytopes.
Publisher
Springer Science and Business Media LLC
Subject
Computational Theory and Mathematics,Discrete Mathematics and Combinatorics,Geometry and Topology,Theoretical Computer Science
Reference30 articles.
1. Armstrong, D., Garsia, A., Haglund, J., Rhoades, B., Sagan, B.: Combinatorics of Tesler matrices in the theory of parking functions and diagonal harmonics. J. Comb. 3(3), 451–494 (2012)
2. Zurich Lectures in Advanced Mathematics;A Barvinok,2008
3. Berline, N., Vergne, M.: Local Euler–Maclaurin formula for polytopes. Mosc. Math. J. 7(3), 355–386 (2007)
4. Canfield, E.R., McKay, B.D.: The asymptotic volume of the Birkhoff polytope. Online J. Anal. Comb. 4, # 2 (2009)
5. Castillo, F., Liu, F.: Berline–Vergne valuation and generalized permutohedra (2015). arXiv:1509.07884