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

More bijections for Entringer and Arnold families

Heesung Shin 1, and  Jiang Zeng 2,

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  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}) $.

Keywords: Euler numbers, Springer numbers, Entringer numbers, Arnold numbers, André permutations, Simsun permutations, increasing 1-2 trees.
Mathematics Subject Classification: 05A05, 05A15, 05A19.
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

show all references

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]
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Figure 2.  Sixteen type B increasing 1-2 trees on [3]
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Figure 3.  Bijections between Entringer families and Arnold families
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Table 1.  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
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$ 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
Table 2.  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 $
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$ 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 $
Table 3.  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\} $
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$ 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\} $
Table 4.  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\} $
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$ 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\} $
Table 5.  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 $
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$ 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 $
Table 6.  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 $
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$ \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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