Special Issues

More bijections for Entringer and Arnold families

  • Received: 01 May 2020 Revised: 01 August 2020 Published: 19 October 2020
  • 05A05, 05A15, 05A19

  • The Euler number $ E_n $ (resp. Entringer number $ E_{n,k} $) enumerates the alternating (down-up) permutations of $ \{1,\dots,n\} $ (resp. starting with $ k $). The Springer number $ S_n $ (resp. Arnold number $ S_{n,k} $) enumerates the type $ B $ alternating permutations (resp. starting with $ k $). In this paper, using bijections we first derive the counterparts in André permutations and Simsun permutations for the Entringer numbers $ (E_{n,k}) $, and then the counterparts in signed André permutations and type $ B $ increasing 1-2 trees for the Arnold numbers $ (S_{n,k}) $.

    Citation: Heesung Shin, Jiang Zeng. More bijections for Entringer and Arnold families[J]. Electronic Research Archive, 2021, 29(2): 2167-2185. doi: 10.3934/era.2020111

    Related Papers:

  • The Euler number $ E_n $ (resp. Entringer number $ E_{n,k} $) enumerates the alternating (down-up) permutations of $ \{1,\dots,n\} $ (resp. starting with $ k $). The Springer number $ S_n $ (resp. Arnold number $ S_{n,k} $) enumerates the type $ B $ alternating permutations (resp. starting with $ k $). In this paper, using bijections we first derive the counterparts in André permutations and Simsun permutations for the Entringer numbers $ (E_{n,k}) $, and then the counterparts in signed André permutations and type $ B $ increasing 1-2 trees for the Arnold numbers $ (S_{n,k}) $.



    加载中


    [1] Développement de $\sec x$ et $\tan x$. C. R. Math. Acad. Sci. Paris (1879) 88: 965-979.
    [2] Snake calculus and the combinatorics of the Bernoulli, Euler and Springer numbers of Coxeter groups. Uspekhi Mat. Nauk (1992) 47: 3-45.
    [3] D. Callan, A note on downup permutations and increasing 0-1-2 trees, preprint, http://www.stat.wisc.edu/ callan/papersother/.
    [4] On simsun and double simsun permutations avoiding a pattern of length three. Fund. Inform. (2012) 117: 155-177.
    [5] Alternating permutations and binary increasing trees. J. Combinatorial Theory Ser. A (1975) 18: 141-148.
    [6] Coproducts and the $cd$-index. J. Algebraic Combin. (1998) 8: 273-299.
    [7] A combinatorial interpretation of the {E}uler and Bernoulli numbers. Nieuw Arch. Wisk. (3) (1966) 14: 241-246.
    [8] D. Foata and G.-N. Han, André permutation calculus: A twin Seidel matrix sequence, Sém. Lothar. Combin., 73 ([2014-2016]), Art. B73e, 54 pp.
    [9] D. Foata and M.-P. Schützenberger, Nombres d'euler et permutations alternantes, Manuscript, 71 pages, University of Florida, Gainesville, http://www.mat.univie.ac.at/ slc/, Available in the 'Books' section of the Séminaire Lotharingien de Combinatoire. doi: 10.1016/B978-0-7204-2262-7.50021-1
    [10] D. Foata and M.-P. Schützenberger, Nombres d'Euler et permutations alternantes, A Survey of Combinatorial Theory, North-Holland, Amsterdam, (1973), 173–187.
    [11] Bijections for Entringer families. European J. Combin. (2011) 32: 100-115.
    [12] On the $cd$-variation polynomials of André and Simsun permutations. Discrete Comput. Geom. (1996) 16: 259-275.
    [13] The algebraic combinatorics of snakes. J. Combin. Theory Ser. A (2012) 119: 1613-1638.
    [14] On the shape of polynomial curves. Tohoku Mathematical J. (1933) 37: 347-362.
    [15] De nouvelles significations énumératives des nombres d'Entringer. Discrete Math. (1982) 38: 265-271.
    [16] Deux propriétés des arbres binaires ordonnés stricts. European J. Combin. (1989) 10: 369-374.
    [17] Two other interpretations of the Entringer numbers. European J. Combin. (1997) 18: 939-943.
    [18] André permutations, lexicographic shellability and the $cd$-index of a convex polytope. Trans. Amer. Math. Soc. (1993) 338: 77-104.
    [19] Über eine einfache entstehungsweise der bernoullischen zahlen und einiger verwandten reihen. Sitzungsber. Münch. Akad. (1877) 4: 157-187.
    [20] The on-line encyclopedia of integer sequences. Notices Amer. Math. Soc. (2018) 65: 1062-1074.
    [21] Remarks on a combinatorial problem. Nieuw Arch. Wisk. (3) (1971) 19: 30-36.
    [22] Flag $f$-vectors and the $cd$-index. Math. Z. (1994) 216: 483-499.
    [23] A survey of alternating permutations. Combinatorics and Graphs, Contemp. Math., Amer. Math. Soc., Providence, RI (2010) 531: 165-196.
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1931) PDF downloads(204) Cited by(2)

Article outline

Figures and Tables

Figures(3)  /  Tables(6)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog