Geodesic
Flow Polytopes and the Space of Diagonal Harmonics
Canadian journal of mathematics, Tome 71 (2019) no. 6, pp. 1495-1521

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DOI

A result of Haglund implies that the $(q,t)$-bigraded Hilbert series of the space of diagonal harmonics is a $(q,t)$-Ehrhart function of the flow polytope of a complete graph with netflow vector $(-n,1,\ldots ,1)$. We study the $(q,t)$-Ehrhart functions of flow polytopes of threshold graphs with arbitrary netflow vectors. Our results generalize previously known specializations of the mentioned bigraded Hilbert series at $t=1$, $0$, and $q^{-1}$. As a corollary to our results, we obtain a proof of a conjecture of Armstrong, Garsia, Haglund, Rhoades, and Sagan about the $(q,q^{-1})$-Ehrhart function of the flow polytope of a complete graph with an arbitrary netflow vector.
DOI : 10.4153/CJM-2018-007-3
Mots-clés : flow polytope, threshold graph, diagonal harmonic, Tesler matrix 
Liu, Ricky Ini; Morales, Alejandro H.; Mészáros, Karola. Flow Polytopes and the Space of Diagonal Harmonics. Canadian journal of mathematics, Tome 71 (2019) no. 6, pp. 1495-1521. doi: 10.4153/CJM-2018-007-3
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     title = {Flow {Polytopes} and the {Space} of {Diagonal} {Harmonics}},
     journal = {Canadian journal of mathematics},
     pages = {1495--1521},
     year = {2019},
     volume = {71},
     number = {6},
     doi = {10.4153/CJM-2018-007-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2018-007-3/}
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