Lattice point generating functions and symmetric cones

Matthias Beck; Thomas Bliem; Benjamin Braun; Carla D. Savage

doi:10.1007/s10801-012-0414-9

Lattice point generating functions and symmetric cones

Matthias Beck, Thomas Bliem, Benjamin Braun, Carla D. Savage

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We show that a recent identity of Beck-Gessel-Lee-Savage on the generating function of symmetrically constrained compositions of integers generalizes naturally to a family of convex polyhedral cones that are invariant under the action of a finite reflection group. We obtain general expressions for the multivariate generating functions of such cones, and work out their general form more specifically for all symmetry groups of type A (previously known) and types B and D (new). We obtain several applications of these expressions in type B, including identities involving permutation statistics and lecture hall partitions.

Original language	English
Pages (from-to)	543-566
Number of pages	24
Journal	Journal of Algebraic Combinatorics
Volume	38
Issue number	3
DOIs	https://doi.org/10.1007/s10801-012-0414-9
State	Published - Nov 2013

Bibliographical note

Funding Information:
Acknowledgements We thank the anonymous referees for their thoughtful comments. M.B. is partially supported by the NSF (DMS-0810105 & DMS-1162638). T.B. has been supported by the Deutsche Forschungsgemeinschaft (SPP 1388). B.B. is partially supported by the NSF (DMS-0758321). We are grateful to the American Institute of Mathematics for supporting our SQuaRE working group “Polyhedral Geometry and Partition Theory” where our collaboration on this work began.

Funding

Acknowledgements We thank the anonymous referees for their thoughtful comments. M.B. is partially supported by the NSF (DMS-0810105 & DMS-1162638). T.B. has been supported by the Deutsche Forschungsgemeinschaft (SPP 1388). B.B. is partially supported by the NSF (DMS-0758321). We are grateful to the American Institute of Mathematics for supporting our SQuaRE working group “Polyhedral Geometry and Partition Theory” where our collaboration on this work began.

Funders	Funder number
National Science Foundation Arctic Social Science Program	DMS-0810105, DMS-1162638
National Science Foundation Arctic Social Science Program
Directorate for Mathematical and Physical Sciences	0758321, 0810105
Directorate for Mathematical and Physical Sciences
Deutsche Forschungsgemeinschaft	DMS-0758321, SPP 1388
Deutsche Forschungsgemeinschaft

Keywords

Coxeter group
Finite reflection group
Lattice point generating function
Lecture hall partition
Permutation statistics
Polyhedral cone
Symmetrically constrained composition

ASJC Scopus subject areas

Algebra and Number Theory
Discrete Mathematics and Combinatorics

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10.1007/s10801-012-0414-9

Cite this

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abstract = "We show that a recent identity of Beck-Gessel-Lee-Savage on the generating function of symmetrically constrained compositions of integers generalizes naturally to a family of convex polyhedral cones that are invariant under the action of a finite reflection group. We obtain general expressions for the multivariate generating functions of such cones, and work out their general form more specifically for all symmetry groups of type A (previously known) and types B and D (new). We obtain several applications of these expressions in type B, including identities involving permutation statistics and lecture hall partitions.",

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note = "Funding Information: Acknowledgements We thank the anonymous referees for their thoughtful comments. M.B. is partially supported by the NSF (DMS-0810105 & DMS-1162638). T.B. has been supported by the Deutsche Forschungsgemeinschaft (SPP 1388). B.B. is partially supported by the NSF (DMS-0758321). We are grateful to the American Institute of Mathematics for supporting our SQuaRE working group “Polyhedral Geometry and Partition Theory” where our collaboration on this work began.",

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N2 - We show that a recent identity of Beck-Gessel-Lee-Savage on the generating function of symmetrically constrained compositions of integers generalizes naturally to a family of convex polyhedral cones that are invariant under the action of a finite reflection group. We obtain general expressions for the multivariate generating functions of such cones, and work out their general form more specifically for all symmetry groups of type A (previously known) and types B and D (new). We obtain several applications of these expressions in type B, including identities involving permutation statistics and lecture hall partitions.

AB - We show that a recent identity of Beck-Gessel-Lee-Savage on the generating function of symmetrically constrained compositions of integers generalizes naturally to a family of convex polyhedral cones that are invariant under the action of a finite reflection group. We obtain general expressions for the multivariate generating functions of such cones, and work out their general form more specifically for all symmetry groups of type A (previously known) and types B and D (new). We obtain several applications of these expressions in type B, including identities involving permutation statistics and lecture hall partitions.

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KW - Permutation statistics

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Lattice point generating functions and symmetric cones

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

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