Descent pattern avoidance

Richard Ehrenborg; Jiyoon Jung

doi:10.1016/j.aam.2012.08.004

Descent pattern avoidance

Richard Ehrenborg, Jiyoon Jung

Mathematics

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

7 Scopus citations

Abstract

We extend the notion of consecutive pattern avoidance to considering sums over all permutations where each term is a product of weights depending on each consecutive pattern of a fixed length. We study the problem of finding the asymptotics of these sums. Our technique is to extend the spectral method of Ehrenborg, Kitaev and Perry. When the weight depends on the descent pattern we show how to find the equation determining the spectrum. We give two length 4 applications. First, we find the asymptotics of the number of permutations with no triple ascents and no triple descents. Second, we give the asymptotics of the number of permutations with no isolated ascents or descents. Our next result is a weighted pattern of length 3 where the associated operator only has one non-zero eigenvalue. Using generating functions we show that the error term in the asymptotic expression is the smallest possible.

Original language	English
Pages (from-to)	375-390
Number of pages	16
Journal	Advances in Applied Mathematics
Volume	49
Issue number	3-5
DOIs	https://doi.org/10.1016/j.aam.2012.08.004
State	Published - Sep 2012

Bibliographical note

Funding Information:
The authors thank Margaret Readdy and the two referees for their comments on an earlier draft of this paper. The authors are partially funded by the National Science Foundation grant DMS-0902063. The first author also thanks the Institute for Advanced Study and is also partially supported by the National Science Foundation grants DMS-0835373 and CCF-0832797.

Funding

The authors thank Margaret Readdy and the two referees for their comments on an earlier draft of this paper. The authors are partially funded by the National Science Foundation grant DMS-0902063. The first author also thanks the Institute for Advanced Study and is also partially supported by the National Science Foundation grants DMS-0835373 and CCF-0832797.

Funders	Funder number
National Science Foundation (NSF)	0835373, 0902063, DMS-0902063, 0832797
Institute for Advanced Study	CCF-0832797, DMS-0835373

Keywords

Asymptotic expansions
Integral operators
Spectrum
Weighted consecutive pattern avoidance

ASJC Scopus subject areas

Applied Mathematics

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10.1016/j.aam.2012.08.004

Cite this

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abstract = "We extend the notion of consecutive pattern avoidance to considering sums over all permutations where each term is a product of weights depending on each consecutive pattern of a fixed length. We study the problem of finding the asymptotics of these sums. Our technique is to extend the spectral method of Ehrenborg, Kitaev and Perry. When the weight depends on the descent pattern we show how to find the equation determining the spectrum. We give two length 4 applications. First, we find the asymptotics of the number of permutations with no triple ascents and no triple descents. Second, we give the asymptotics of the number of permutations with no isolated ascents or descents. Our next result is a weighted pattern of length 3 where the associated operator only has one non-zero eigenvalue. Using generating functions we show that the error term in the asymptotic expression is the smallest possible.",

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author = "Richard Ehrenborg and Jiyoon Jung",

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N2 - We extend the notion of consecutive pattern avoidance to considering sums over all permutations where each term is a product of weights depending on each consecutive pattern of a fixed length. We study the problem of finding the asymptotics of these sums. Our technique is to extend the spectral method of Ehrenborg, Kitaev and Perry. When the weight depends on the descent pattern we show how to find the equation determining the spectrum. We give two length 4 applications. First, we find the asymptotics of the number of permutations with no triple ascents and no triple descents. Second, we give the asymptotics of the number of permutations with no isolated ascents or descents. Our next result is a weighted pattern of length 3 where the associated operator only has one non-zero eigenvalue. Using generating functions we show that the error term in the asymptotic expression is the smallest possible.

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KW - Asymptotic expansions

KW - Integral operators

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Descent pattern avoidance

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

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