Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Springer Numbers and Arnold Families Revisited

Springer Numbers and Arnold Families Revisited For the calculation of Springer numbers (of root systems) of type Bn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B_n$$\end{document} and Dn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$D_n$$\end{document}, Arnold introduced a signed analogue of alternating permutations, called βn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\beta _n$$\end{document}-snakes, and derived recurrence relations for enumerating the βn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\beta _n$$\end{document}-snakes starting with k. The results are presented in the form of double triangular arrays (vn,k\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$v_{n,k}$$\end{document}) of integers, 1≤|k|≤n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$1\le |k|\le n$$\end{document}. An Arnold family is a sequence of sets of such objects as βn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\beta _n$$\end{document}-snakes that are counted by (vn,k)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(v_{n,k})$$\end{document}. As a refinement of Arnold’s result, we give analogous arrays of polynomials, defined by recurrence, for the calculation of the polynomials associated with successive derivatives of tanx\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tan x$$\end{document} and secx\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sec x$$\end{document}, established by Hoffman. Moreover, we provide some new Arnold families of combinatorial objects that realize the polynomial arrays, which are signed variants of André permutations and Simsun permutations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Arnold Mathematical Journal Springer Journals

Springer Numbers and Arnold Families Revisited

Eu, Sen-Peng; Fu, Tung-Shan
Arnold Mathematical Journal , Volume OnlineFirst – Jun 1, 2023
Download PDF
30 pages

Loading next page...
 
/lp/springer-journals/springer-numbers-and-arnold-families-revisited-Owx3H9MWG5
Publisher
Springer Journals
Copyright
Copyright © Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
ISSN
2199-6792
eISSN
2199-6806
DOI
10.1007/s40598-023-00230-9
Publisher site
See Article on Publisher Site

Abstract

For the calculation of Springer numbers (of root systems) of type Bn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B_n$$\end{document} and Dn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$D_n$$\end{document}, Arnold introduced a signed analogue of alternating permutations, called βn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\beta _n$$\end{document}-snakes, and derived recurrence relations for enumerating the βn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\beta _n$$\end{document}-snakes starting with k. The results are presented in the form of double triangular arrays (vn,k\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$v_{n,k}$$\end{document}) of integers, 1≤|k|≤n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$1\le |k|\le n$$\end{document}. An Arnold family is a sequence of sets of such objects as βn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\beta _n$$\end{document}-snakes that are counted by (vn,k)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(v_{n,k})$$\end{document}. As a refinement of Arnold’s result, we give analogous arrays of polynomials, defined by recurrence, for the calculation of the polynomials associated with successive derivatives of tanx\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tan x$$\end{document} and secx\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sec x$$\end{document}, established by Hoffman. Moreover, we provide some new Arnold families of combinatorial objects that realize the polynomial arrays, which are signed variants of André permutations and Simsun permutations.

Journal

Arnold Mathematical Journal – Springer Journals

Published: Jun 1, 2023

Keywords: Euler number; Springer number; Alternating permutation; Signed permutation; Primary 05A19; Secondary 05A05; 05A15

References

  • The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups
    Arnol’d, VI
  • Further triangles of Seidel-Arnold type and continued fractions related to Euler and Springer numbers
    Dumont, D
  • Coproducts and the cd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$cd$$\end{document}-index
    Ehrenborg, R; Readdy, M
  • A combinatorial interpretation of the Euler and Bernoulli numbers
    Entringer, RC
  • Nombres d’Euler et permutations alternantes
    Foata, D; Schützenberger, M-P; Srivastava, JN
  • Bijections for Entringer families
    Gelineau, Y; Shin, H; Zeng, J
  • On the cd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$cd$$\end{document}-variation polynomials of André and Simsun permutations
    Hetyei, G
  • Derivative polynomials, Euler polynomials, and associated integer sequences
    Hoffman, ME
  • Enumeration of snakes and cycle-alternating permutations
    Josuat-Vergès, M
  • The algebraic combinatorics of snakes
    Josuat-Vergès, M; Novelli, J-C; Thibon, J-Y
  • More bijections for Entringer and Arnold families
    Shin, H; Zeng, J
  • Remarks on a combinatorial problem
    Springer, TA