### Electronic Research Archive

2021, Issue 2: 2167-2185. doi: 10.3934/era.2020111
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 $(S_{n,k})$.

 [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.
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沈阳化工大学材料科学与工程学院 沈阳 110142

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