Counting integer points of flow polytopes

Kapoor, Kabir; Mészáros, Karola; Setiabrata, Linus

doi:10.48550/arxiv.1906.05592

Cited by 2 publications

(2 citation statements)

References 10 publications

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“…Moreover, from the theory of flow polytopes there are formulas for K Ar (µ) for µ i ∈ Z ≥0 as a weighted sum of values of Kostant's partition function at weights independent of µ (see Section 5.2). These formulas are due to Lidskii [35] and generalized to K Λ (µ) by Baldoni-Vergne [3] and proved via polytope subdivisions in [31,40].…”

Section: Applications Connections and Future Workmentioning

confidence: 91%

Kostant's partition function and magic multiplex juggling sequences

Benedetti,

Hanusa,

Harris

et al. 2020

Preprint

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Kostant's partition function is a vector partition function that counts the number of ways one can express a weight of a Lie algebra g as a nonnegative integral linear combination of the positive roots of g. Multiplex juggling sequences are generalizations of juggling sequences that specify an initial and terminal configuration of balls and allow for multiple balls at any particular discrete height. Magic multiplex juggling sequences generalize further to include magic balls, which cancel with standard balls when they meet at the same height.In this paper, we establish a combinatorial equivalence between positive roots of a Lie algebra and throws during a juggling sequence. This provides a juggling framework to calculate Kostant's partition functions, and a partition function framework to compute the number of juggling sequences. From this equivalence we provide a broad range of consequences and applications connecting this work to polytopes, posets, positroids, and weight multiplicities.

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Section: Applications Connections and Future Workmentioning

confidence: 91%

Kostant's partition function and magic multiplex juggling sequences

Benedetti,

Hanusa,

Harris

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Such a proof would provide a better understanding of how volumes of flow polytope and Kostant partition functions are refined by the subdivision lemma. See [16] for another recent proof of this more general volume formula. 7.4.…”

Section: 3mentioning

confidence: 95%

Refinements and Symmetries of the Morris identity for volumes of flow polytopes

Morales,

Shi

2021

Preprint

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Flow polytopes are an important class of polytopes in combinatorics whose lattice points and volumes have interesting properties and relations. The Chan-Robbins-Yuen (CRY) polytope is a flow polytope with normalized volume equal to the product of consecutive Catalan numbers. Zeilberger proved this by evaluating the Morris constant term identity, but no combinatorial proof is known. There is a refinement of this formula that splits the largest Catalan number into Narayana numbers, which Mészáros gave an interpretation as the volume of a collection of flow polytopes. We introduce a new refinement of the Morris identity with combinatorial interpretations both in terms of lattice points and volumes of flow polytopes. Our results generalize Mészáros's construction and a recent flow polytope interpretation of the Morris identity by Corteel-Kim-Mészáros. We prove the product formula of our refinement following the strategy of the Baldoni-Vergne proof of the Morris identity. Lastly, we study a symmetry of the Morris identity bijectively using the Danilov-Karzanov-Koshevoy triangulation of flow polytopes and a bijection of Mészáros-Morales-Striker.

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Counting integer points of flow polytopes

Cited by 2 publications

References 10 publications

Kostant's partition function and magic multiplex juggling sequences

Kostant's partition function and magic multiplex juggling sequences

Refinements and Symmetries of the Morris identity for volumes of flow polytopes

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