Counting faces in the extended Shi arrangement

Richard Ehrenborg

doi:10.1016/j.aam.2019.05.002

Counting faces in the extended Shi arrangement

Richard Ehrenborg

Mathematics

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

1 Scopus citations

Abstract

We define the level of a subset X of Euclidean space to be the dimension of the smallest subspace such that the distance between each element of X and the subspace is bounded. We prove that the number of faces in the n-dimensional extended Shi arrangement Aˆ_n ^r having codimension k and level m is given by m⋅(nk)⋅Δ_r ^kΔ^m−1xⁿ⁻¹|_x=rn−1, where Δ is the difference operator and Δ_r is the difference operator of step r, that is, Δ_rp(x)=p(x)−p(x−r). This generalizes a result of Athanasiadis which counts the number of faces of different dimensions in the extended Shi arrangement Aˆ_n ^r. The proof relies on a multi-variated Abel identity due to Françon.

Original language	English
Pages (from-to)	55-64
Number of pages	10
Journal	Advances in Applied Mathematics
Volume	109
DOIs	https://doi.org/10.1016/j.aam.2019.05.002
State	Published - Aug 2019

Bibliographical note

Funding Information:
The author thanks the referee for bringing the dissertation of Athanasiadis to his attention. The author also thanks Margaret Readdy for inspiring discussions and Richard Stanley and Thomas Zaslavsky for bringing the extended Shi arrangement to the author's attention. The author thanks Jennifer Rice for comments on an earlier version of this article. This research was partially supported by Swedish Natural Science Research Council DNR 702-238/98 . The author was also supported by National Science Foundation grants DMS 98-00910 and DMS 99-83660 while visiting Cornell University. This work was partially supported by a grant from the Simons Foundation (# 429370 to Richard Ehrenborg).

Publisher Copyright:
© 2019 Elsevier Inc.

ASJC Scopus subject areas

Applied Mathematics

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10.1016/j.aam.2019.05.002

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title = "Counting faces in the extended Shi arrangement",

abstract = "We define the level of a subset X of Euclidean space to be the dimension of the smallest subspace such that the distance between each element of X and the subspace is bounded. We prove that the number of faces in the n-dimensional extended Shi arrangement Aˆn r having codimension k and level m is given by m⋅(nk)⋅Δr kΔm−1xn−1|x=rn−1, where Δ is the difference operator and Δr is the difference operator of step r, that is, Δrp(x)=p(x)−p(x−r). This generalizes a result of Athanasiadis which counts the number of faces of different dimensions in the extended Shi arrangement Aˆn r. The proof relies on a multi-variated Abel identity due to Fran{\c c}on.",

author = "Richard Ehrenborg",

note = "Funding Information: The author thanks the referee for bringing the dissertation of Athanasiadis to his attention. The author also thanks Margaret Readdy for inspiring discussions and Richard Stanley and Thomas Zaslavsky for bringing the extended Shi arrangement to the author's attention. The author thanks Jennifer Rice for comments on an earlier version of this article. This research was partially supported by Swedish Natural Science Research Council DNR 702-238/98 . The author was also supported by National Science Foundation grants DMS 98-00910 and DMS 99-83660 while visiting Cornell University. This work was partially supported by a grant from the Simons Foundation (# 429370 to Richard Ehrenborg). Publisher Copyright: {\textcopyright} 2019 Elsevier Inc.",

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N1 - Funding Information: The author thanks the referee for bringing the dissertation of Athanasiadis to his attention. The author also thanks Margaret Readdy for inspiring discussions and Richard Stanley and Thomas Zaslavsky for bringing the extended Shi arrangement to the author's attention. The author thanks Jennifer Rice for comments on an earlier version of this article. This research was partially supported by Swedish Natural Science Research Council DNR 702-238/98 . The author was also supported by National Science Foundation grants DMS 98-00910 and DMS 99-83660 while visiting Cornell University. This work was partially supported by a grant from the Simons Foundation (# 429370 to Richard Ehrenborg). Publisher Copyright: © 2019 Elsevier Inc.

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N2 - We define the level of a subset X of Euclidean space to be the dimension of the smallest subspace such that the distance between each element of X and the subspace is bounded. We prove that the number of faces in the n-dimensional extended Shi arrangement Aˆn r having codimension k and level m is given by m⋅(nk)⋅Δr kΔm−1xn−1|x=rn−1, where Δ is the difference operator and Δr is the difference operator of step r, that is, Δrp(x)=p(x)−p(x−r). This generalizes a result of Athanasiadis which counts the number of faces of different dimensions in the extended Shi arrangement Aˆn r. The proof relies on a multi-variated Abel identity due to Françon.

AB - We define the level of a subset X of Euclidean space to be the dimension of the smallest subspace such that the distance between each element of X and the subspace is bounded. We prove that the number of faces in the n-dimensional extended Shi arrangement Aˆn r having codimension k and level m is given by m⋅(nk)⋅Δr kΔm−1xn−1|x=rn−1, where Δ is the difference operator and Δr is the difference operator of step r, that is, Δrp(x)=p(x)−p(x−r). This generalizes a result of Athanasiadis which counts the number of faces of different dimensions in the extended Shi arrangement Aˆn r. The proof relies on a multi-variated Abel identity due to Françon.

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JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

ER -

Counting faces in the extended Shi arrangement

Abstract

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