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Asymptotics of principal evaluations of Schubert polynomials for layered permutations


Authors: Alejandro H. Morales, Igor Pak and Greta Panova
Journal: Proc. Amer. Math. Soc. 147 (2019), 1377-1389
MSC (2010): Primary 05A05, 05A16, 05E05, 14N15
DOI: https://doi.org/10.1090/proc/14369
Published electronically: January 9, 2019
MathSciNet review: 3910405
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Abstract: Denote by $u(n)$ the largest principal specialization of the Schubert polynomial \begin{equation*} u(n) := \max _{w \in S_n} \hskip .06cm \mathfrak {S}_w(1,\ldots ,1). \end{equation*} Stanley conjectured that there is a limit \begin{equation*} \lim _{n\to \infty } \frac {1}{n^2} \hskip .06cm \log u(n), \end{equation*} and asked for a limiting description of permutations achieving the maximum $u(n)$. Merzon and Smirnov conjectured in [Eur. J. Math. 2 (2016), pp. 227–245] that this maximum is achieved on layered permutations. We resolve both of Stanley’s problems restricted to layered permutations.


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Additional Information

Alejandro H. Morales
Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003
MR Author ID: 819004
Email: ahmorales@math.umass.edu

Igor Pak
Affiliation: Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095
MR Author ID: 293184
ORCID: 0000-0001-8579-7239
Email: pak@math.ucla.edu

Greta Panova
Affiliation: Institute of Advanced Studies, Princeton, New Jersey 08540 –and– Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
Address at time of publication: Department of Mathematics, University of Southern California, Los Angeles, California 90089
MR Author ID: 964307
Email: gpanova@usc.edu

Received by editor(s): May 29, 2018
Received by editor(s) in revised form: June 27, 2018
Published electronically: January 9, 2019
Additional Notes: The second and third authors were partially supported by the NSF
Communicated by: Patricia L. Hersh
Article copyright: © Copyright 2019 American Mathematical Society