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June  2021, 29(2): 2167-2185. doi: 10.3934/era.2020111

## More bijections for Entringer and Arnold families

 1 Department of Mathematics, Inha University, Incheon 22212, Korea 2 Université de Lyon; Université Lyon 1; UMR 5208 du CNRS, Institut Camille Jordan, 43, boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France

Received  May 2020 Revised  August 2020 Published  June 2021 Early access  October 2020

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. Electronic Research Archive, 2021, 29 (2) : 2167-2185. doi: 10.3934/era.2020111
##### References:
 [1] D. André, Développement de $\sec x$ et $\tan x$, C. R. Math. Acad. Sci. Paris, 88 (1879), 965-979.   Google Scholar [2] V. I. Arnol'd, Snake calculus and the combinatorics of the Bernoulli, Euler and Springer numbers of Coxeter groups, Uspekhi Mat. Nauk, 47 (1992), 3-45.  doi: 10.1070/RM1992v047n01ABEH000861.  Google Scholar [3] D. Callan, A note on downup permutations and increasing 0-1-2 trees, preprint, http://www.stat.wisc.edu/ callan/papersother/. Google Scholar [4] W.-C. Chuang, S.-P. Eu, T.-S. Fu and Y.-J. Pan, On simsun and double simsun permutations avoiding a pattern of length three, Fund. Inform., 117 (2012), 155-177.  doi: 10.3233/FI-2012-693.  Google Scholar [5] R. Donaghey, Alternating permutations and binary increasing trees, J. Combinatorial Theory Ser. A, 18 (1975), 141-148.  doi: 10.1016/0097-3165(75)90002-3.  Google Scholar [6] R. Ehrenborg and M. Readdy, Coproducts and the $cd$-index, J. Algebraic Combin., 8 (1998), 273-299.  doi: 10.1023/A:1008614816374.  Google Scholar [7] R. C. Entringer, A combinatorial interpretation of the {E}uler and Bernoulli numbers, Nieuw Arch. Wisk. (3), 14 (1966), 241-246.   Google Scholar [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.  Google Scholar [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.  Google Scholar [10] D. Foata and M.-P. Schützenberger, Nombres d'Euler et permutations alternantes, A Survey of Combinatorial Theory, North-Holland, Amsterdam, (1973), 173–187.  Google Scholar [11] Y. Gelineau, H. Shin and J. Zeng, Bijections for Entringer families, European J. Combin., 32 (2011), 100-115.  doi: 10.1016/j.ejc.2010.07.004.  Google Scholar [12] G. Hetyei, On the $cd$-variation polynomials of André and Simsun permutations, Discrete Comput. Geom., 16 (1996), 259-275.  doi: 10.1007/BF02711512.  Google Scholar [13] M. Josuat-Vergès, J.-C. Novelli and J.-Y. Thibon, The algebraic combinatorics of snakes, J. Combin. Theory Ser. A, 119 (2012), 1613-1638.  doi: 10.1016/j.jcta.2012.05.002.  Google Scholar [14] A. J. Kempner, On the shape of polynomial curves, Tohoku Mathematical J., 37 (1933), 347-362.   Google Scholar [15] C. Poupard, De nouvelles significations énumératives des nombres d'Entringer, Discrete Math., 38 (1982), 265-271.  doi: 10.1016/0012-365X(82)90293-X.  Google Scholar [16] C. Poupard, Deux propriétés des arbres binaires ordonnés stricts, European J. Combin., 10 (1989), 369-374.  doi: 10.1016/S0195-6698(89)80009-5.  Google Scholar [17] C. Poupard, Two other interpretations of the Entringer numbers, European J. Combin., 18 (1997), 939-943.  doi: 10.1006/eujc.1997.0147.  Google Scholar [18] M. Purtill, André permutations, lexicographic shellability and the $cd$-index of a convex polytope, Trans. Amer. Math. Soc., 338 (1993), 77-104.  doi: 10.2307/2154445.  Google Scholar [19] P. L. v. Seidel, Über eine einfache entstehungsweise der bernoullischen zahlen und einiger verwandten reihen, Sitzungsber. Münch. Akad., 4 (1877), 157-187.   Google Scholar [20] N. J. A. Sloane, The on-line encyclopedia of integer sequences, Notices Amer. Math. Soc., 65 (2018), 1062-1074.   Google Scholar [21] T. A. Springer, Remarks on a combinatorial problem, Nieuw Arch. Wisk. (3), 19 (1971), 30-36.   Google Scholar [22] R. P. Stanley, Flag $f$-vectors and the $cd$-index, Math. Z., 216 (1994), 483-499.  doi: 10.1007/BF02572336.  Google Scholar [23] R. P. Stanley, A survey of alternating permutations, Combinatorics and Graphs, Contemp. Math., Amer. Math. Soc., Providence, RI, 531 (2010), 165-196.  doi: 10.1090/conm/531/10466.  Google Scholar

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##### References:
 [1] D. André, Développement de $\sec x$ et $\tan x$, C. R. Math. Acad. Sci. Paris, 88 (1879), 965-979.   Google Scholar [2] V. I. Arnol'd, Snake calculus and the combinatorics of the Bernoulli, Euler and Springer numbers of Coxeter groups, Uspekhi Mat. Nauk, 47 (1992), 3-45.  doi: 10.1070/RM1992v047n01ABEH000861.  Google Scholar [3] D. Callan, A note on downup permutations and increasing 0-1-2 trees, preprint, http://www.stat.wisc.edu/ callan/papersother/. Google Scholar [4] W.-C. Chuang, S.-P. Eu, T.-S. Fu and Y.-J. Pan, On simsun and double simsun permutations avoiding a pattern of length three, Fund. Inform., 117 (2012), 155-177.  doi: 10.3233/FI-2012-693.  Google Scholar [5] R. Donaghey, Alternating permutations and binary increasing trees, J. Combinatorial Theory Ser. A, 18 (1975), 141-148.  doi: 10.1016/0097-3165(75)90002-3.  Google Scholar [6] R. Ehrenborg and M. Readdy, Coproducts and the $cd$-index, J. Algebraic Combin., 8 (1998), 273-299.  doi: 10.1023/A:1008614816374.  Google Scholar [7] R. C. Entringer, A combinatorial interpretation of the {E}uler and Bernoulli numbers, Nieuw Arch. Wisk. (3), 14 (1966), 241-246.   Google Scholar [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.  Google Scholar [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.  Google Scholar [10] D. Foata and M.-P. Schützenberger, Nombres d'Euler et permutations alternantes, A Survey of Combinatorial Theory, North-Holland, Amsterdam, (1973), 173–187.  Google Scholar [11] Y. Gelineau, H. Shin and J. Zeng, Bijections for Entringer families, European J. Combin., 32 (2011), 100-115.  doi: 10.1016/j.ejc.2010.07.004.  Google Scholar [12] G. Hetyei, On the $cd$-variation polynomials of André and Simsun permutations, Discrete Comput. Geom., 16 (1996), 259-275.  doi: 10.1007/BF02711512.  Google Scholar [13] M. Josuat-Vergès, J.-C. Novelli and J.-Y. Thibon, The algebraic combinatorics of snakes, J. Combin. Theory Ser. A, 119 (2012), 1613-1638.  doi: 10.1016/j.jcta.2012.05.002.  Google Scholar [14] A. J. Kempner, On the shape of polynomial curves, Tohoku Mathematical J., 37 (1933), 347-362.   Google Scholar [15] C. Poupard, De nouvelles significations énumératives des nombres d'Entringer, Discrete Math., 38 (1982), 265-271.  doi: 10.1016/0012-365X(82)90293-X.  Google Scholar [16] C. Poupard, Deux propriétés des arbres binaires ordonnés stricts, European J. Combin., 10 (1989), 369-374.  doi: 10.1016/S0195-6698(89)80009-5.  Google Scholar [17] C. Poupard, Two other interpretations of the Entringer numbers, European J. Combin., 18 (1997), 939-943.  doi: 10.1006/eujc.1997.0147.  Google Scholar [18] M. Purtill, André permutations, lexicographic shellability and the $cd$-index of a convex polytope, Trans. Amer. Math. Soc., 338 (1993), 77-104.  doi: 10.2307/2154445.  Google Scholar [19] P. L. v. Seidel, Über eine einfache entstehungsweise der bernoullischen zahlen und einiger verwandten reihen, Sitzungsber. Münch. Akad., 4 (1877), 157-187.   Google Scholar [20] N. J. A. Sloane, The on-line encyclopedia of integer sequences, Notices Amer. Math. Soc., 65 (2018), 1062-1074.   Google Scholar [21] T. A. Springer, Remarks on a combinatorial problem, Nieuw Arch. Wisk. (3), 19 (1971), 30-36.   Google Scholar [22] R. P. Stanley, Flag $f$-vectors and the $cd$-index, Math. Z., 216 (1994), 483-499.  doi: 10.1007/BF02572336.  Google Scholar [23] R. P. Stanley, A survey of alternating permutations, Combinatorics and Graphs, Contemp. Math., Amer. Math. Soc., Providence, RI, 531 (2010), 165-196.  doi: 10.1090/conm/531/10466.  Google Scholar
]">Figure 1.  increasing 1-2 trees on [4]
]">Figure 2.  Sixteen type B increasing 1-2 trees on [3]
Bijections between Entringer families and Arnold families
The Entringer numbers ${{E}_{n,k}}$ for $1\leq k \leq n\leq 7$ and Euler numbers ${{E}_n} = \sum_{k = 1}^n {{E}_{n,k}}$
 $n\setminus k$ 1 2 3 4 5 6 7 $E_n$ 1 1 1 2 0 1 1 3 0 1 1 2 4 0 1 2 2 5 5 0 2 4 5 5 16 6 0 5 10 14 16 16 61 7 0 16 32 46 56 61 61 271
 $n\setminus k$ 1 2 3 4 5 6 7 $E_n$ 1 1 1 2 0 1 1 3 0 1 1 2 4 0 1 2 2 5 5 0 2 4 5 5 16 6 0 5 10 14 16 16 61 7 0 16 32 46 56 61 61 271
The Arnold numbers ${S_{n,k}}$ for $1\leq \left|{k}\right| \leq n\leq 6$ and Springer numbers ${S_n} = \sum _{k = 1}^n{S_{n,k}}$
 $n\setminus k$ -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 $S_n$ 1 1 1 $1$ 2 0 1 1 2 $3$ 3 0 2 3 3 4 4 $11$ 4 0 4 8 11 11 14 16 16 $57$ 5 0 16 32 46 57 57 68 76 80 80 $361$ 6 0 80 160 236 304 361 361 418 464 496 512 512 $2763$
 $n\setminus k$ -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 $S_n$ 1 1 1 $1$ 2 0 1 1 2 $3$ 3 0 2 3 3 4 4 $11$ 4 0 4 8 11 11 14 16 16 $57$ 5 0 16 32 46 57 57 68 76 80 80 $361$ 6 0 80 160 236 304 361 361 418 464 496 512 512 $2763$
The sets ${\mathcal{DU} }_{4,k}$, ${\mathcal{A}}_{4,k}$, and ${\mathcal{RS}}_{3,k-1}$ for $2\le k \le 4$
 $k$ ${\mathcal{DU} }_{4,k}$ ${\mathcal{A}}_{4,k}$ ${\mathcal{RS}}_{3,k-1}$ $2$ $\left\{{2143}\right\}$ $\left\{{3412}\right\}$ $\left\{{231}\right\}$ $3$ $\left\{{3142, 3241}\right\}$ $\left\{{1423, 4123}\right\}$ $\left\{{132,312}\right\}$ $4$ $\left\{{4132, 4231}\right\}$ $\left\{{1234, 3124}\right\}$ $\left\{{123,213}\right\}$
 $k$ ${\mathcal{DU} }_{4,k}$ ${\mathcal{A}}_{4,k}$ ${\mathcal{RS}}_{3,k-1}$ $2$ $\left\{{2143}\right\}$ $\left\{{3412}\right\}$ $\left\{{231}\right\}$ $3$ $\left\{{3142, 3241}\right\}$ $\left\{{1423, 4123}\right\}$ $\left\{{132,312}\right\}$ $4$ $\left\{{4132, 4231}\right\}$ $\left\{{1234, 3124}\right\}$ $\left\{{123,213}\right\}$
The sets ${\mathcal{S}}_{3,k}$, ${{\mathcal{A}}^{(B)}}_{3,k}$, ${{\mathcal{A}}^{(H)}}_{4,5-k}$, and ${{\mathcal{RS}}^{(B)}}_{3, 4-k}$ for $1\le k \le 3$
 $k$ ${\mathcal{S}}_{3,k}$ ${{\mathcal{A}}^{(B)}}_{3,k}$ ${{\mathcal{A}}^{(H)}}_{4,5-k}$ ${{\mathcal{RS}}^{(B)}}_{3,4-k}$ $1$ $\left\{{1\bar{2}3, 1\bar{3}2, 1\bar{3}\bar{2}}\right\}$ $\left\{{3\bar{2}1, \bar{3}\bar{2}1, 2\bar{3}1}\right\}$ $\left\{{1234, 3124, \bar{3}124}\right\}$ $\left\{{123,213, \bar{2}13}\right\}$ $2$ $\left\{ 213, 2\bar{1}3, \qquad \right. \left. \qquad 2\bar{3}1, 2\bar{3}\bar{1} \right\}$ $\left\{ 312, \bar{3}12, \qquad \right. \left. \qquad 3\bar{1}2, \bar{3}\bar{1}2 \right\}$ $\left\{1423, 1\bar{4}23, \qquad \right. \left. \qquad 4123, \bar{4}123 \right\}$ $\left\{132, 1\bar{3}2, \qquad \right. \left. \qquad 312, \bar{3}12 \right\}$ $3$ $\left\{312, 3\bar{1}2, \qquad \right. \left. \qquad 3\bar{2}1, 3\bar{2}\bar{1} \right\}$ $\left\{\bar{2}13, \bar{2}\bar{1}3, \qquad \right. \left. \qquad 123, \bar{1}23 \right\}$ $\left\{3412, \bar{3}412, \qquad \right. \left. \qquad 3\bar{4}12, \bar{3}\bar{4}12 \right\}$ $\left\{231, \bar{2}31, \qquad \right. \left. \qquad 2\bar{3}1, \bar{2}\bar{3}1 \right\}$
 $k$ ${\mathcal{S}}_{3,k}$ ${{\mathcal{A}}^{(B)}}_{3,k}$ ${{\mathcal{A}}^{(H)}}_{4,5-k}$ ${{\mathcal{RS}}^{(B)}}_{3,4-k}$ $1$ $\left\{{1\bar{2}3, 1\bar{3}2, 1\bar{3}\bar{2}}\right\}$ $\left\{{3\bar{2}1, \bar{3}\bar{2}1, 2\bar{3}1}\right\}$ $\left\{{1234, 3124, \bar{3}124}\right\}$ $\left\{{123,213, \bar{2}13}\right\}$ $2$ $\left\{ 213, 2\bar{1}3, \qquad \right. \left. \qquad 2\bar{3}1, 2\bar{3}\bar{1} \right\}$ $\left\{ 312, \bar{3}12, \qquad \right. \left. \qquad 3\bar{1}2, \bar{3}\bar{1}2 \right\}$ $\left\{1423, 1\bar{4}23, \qquad \right. \left. \qquad 4123, \bar{4}123 \right\}$ $\left\{132, 1\bar{3}2, \qquad \right. \left. \qquad 312, \bar{3}12 \right\}$ $3$ $\left\{312, 3\bar{1}2, \qquad \right. \left. \qquad 3\bar{2}1, 3\bar{2}\bar{1} \right\}$ $\left\{\bar{2}13, \bar{2}\bar{1}3, \qquad \right. \left. \qquad 123, \bar{1}23 \right\}$ $\left\{3412, \bar{3}412, \qquad \right. \left. \qquad 3\bar{4}12, \bar{3}\bar{4}12 \right\}$ $\left\{231, \bar{2}31, \qquad \right. \left. \qquad 2\bar{3}1, \bar{2}\bar{3}1 \right\}$
Three bijections between Entringer families with $n = 4$ and $2\le k \le 4$
 $k$ $\tau \in {\mathcal{DU} }_{4,k}$ $\psi(\tau) \in {\mathcal{T}}_{4,k}$ $\omega(\psi(\tau)) \in {\mathcal{A}}_{4,k}$ $\varphi(\omega(\psi(\tau))) \in {\mathcal{RS}}_{3,k-1}$ $2$ $2143$ $3412$ $231$ $3$ $3241$ $1423$ $132$ $3142$ $4123$ $312$ $4$ $4231$ $1234$ $123$ $4132$ $3124$ $213$
 $k$ $\tau \in {\mathcal{DU} }_{4,k}$ $\psi(\tau) \in {\mathcal{T}}_{4,k}$ $\omega(\psi(\tau)) \in {\mathcal{A}}_{4,k}$ $\varphi(\omega(\psi(\tau))) \in {\mathcal{RS}}_{3,k-1}$ $2$ $2143$ $3412$ $231$ $3$ $3241$ $1423$ $132$ $3142$ $4123$ $312$ $4$ $4231$ $1234$ $123$ $4132$ $3124$ $213$
Three bijections between Arnold families with $n = 3$ and $1\le k \le 3$
 $\tau \in {\mathcal{S}}_{3,k}$ $\psi^{(B)}(\tau) \in {{\mathcal{T}}^{(B)}}_{3,k}$ $\omega^{(B)}(\psi^{(B)}(\tau)) \in {{\mathcal{A}}^{(B)}}_{3,k}$ $k$ $\sigma \in {{\mathcal{A}}^{(H)}}_{4,5-k}$ $\varphi^{(B)}(\sigma) \in {{\mathcal{RS}}^{(B)}}_{3,4-k}$ $1\bar{2}3$ $3\bar{2}1$ $1$ $1234$ $123$ $1\bar{3}2$ $2\bar{3}1$ $3124$ $213$ $1\bar{3}\bar{2}$ $\bar{3}\bar{2}1$ $\bar{3}124$ $\bar{2}13$ $213$ $312$ $2$ $1423$ $132$ $2\bar{1}3$ $3\bar{1}2$ $1\bar{4}23$ $1\bar{3}2$ $2\bar{3}1$ $\bar{3}12$ $4123$ $312$ $2\bar{3}\bar{1}$ $\bar{3}\bar{1}2$ $\bar{4}123$ $\bar{3}12$ $312$ $123$ $3$ $3412$ $231$ $3\bar{1}2$ $\bar{1}23$ $\bar{3}412$ $\bar{2}31$ $3\bar{2}1$ $\bar{2}13$ $3\bar{4}12$ $2\bar{3}1$ $3\bar{2}\bar{1}$ $\bar{2}\bar{1}3$ $\bar{3}\bar{4}12$ $\bar{2}\bar{3}1$
 $\tau \in {\mathcal{S}}_{3,k}$ $\psi^{(B)}(\tau) \in {{\mathcal{T}}^{(B)}}_{3,k}$ $\omega^{(B)}(\psi^{(B)}(\tau)) \in {{\mathcal{A}}^{(B)}}_{3,k}$ $k$ $\sigma \in {{\mathcal{A}}^{(H)}}_{4,5-k}$ $\varphi^{(B)}(\sigma) \in {{\mathcal{RS}}^{(B)}}_{3,4-k}$ $1\bar{2}3$ $3\bar{2}1$ $1$ $1234$ $123$ $1\bar{3}2$ $2\bar{3}1$ $3124$ $213$ $1\bar{3}\bar{2}$ $\bar{3}\bar{2}1$ $\bar{3}124$ $\bar{2}13$ $213$ $312$ $2$ $1423$ $132$ $2\bar{1}3$ $3\bar{1}2$ $1\bar{4}23$ $1\bar{3}2$ $2\bar{3}1$ $\bar{3}12$ $4123$ $312$ $2\bar{3}\bar{1}$ $\bar{3}\bar{1}2$ $\bar{4}123$ $\bar{3}12$ $312$ $123$ $3$ $3412$ $231$ $3\bar{1}2$ $\bar{1}23$ $\bar{3}412$ $\bar{2}31$ $3\bar{2}1$ $\bar{2}13$ $3\bar{4}12$ $2\bar{3}1$ $3\bar{2}\bar{1}$ $\bar{2}\bar{1}3$ $\bar{3}\bar{4}12$ $\bar{2}\bar{3}1$
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