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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 |
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}) $.
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. |
[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. |
[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. |
[6] |
R. Ehrenborg and M. Readdy,
Coproducts and the $cd$-index, J. Algebraic Combin., 8 (1998), 273-299.
doi: 10.1023/A:1008614816374. |
[7] |
R. C. Entringer,
A combinatorial interpretation of the {E}uler and Bernoulli numbers, Nieuw Arch. Wisk. (3), 14 (1966), 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] |
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. |
[12] |
G. Hetyei,
On the $cd$-variation polynomials of André and Simsun permutations, Discrete Comput. Geom., 16 (1996), 259-275.
doi: 10.1007/BF02711512. |
[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. |
[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. |
[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. |
[17] |
C. Poupard,
Two other interpretations of the Entringer numbers, European J. Combin., 18 (1997), 939-943.
doi: 10.1006/eujc.1997.0147. |
[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. |
[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.
|
[21] |
T. A. Springer,
Remarks on a combinatorial problem, Nieuw Arch. Wisk. (3), 19 (1971), 30-36.
|
[22] |
R. P. Stanley,
Flag $f$-vectors and the $cd$-index, Math. Z., 216 (1994), 483-499.
doi: 10.1007/BF02572336. |
[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. |
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. |
[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. |
[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. |
[6] |
R. Ehrenborg and M. Readdy,
Coproducts and the $cd$-index, J. Algebraic Combin., 8 (1998), 273-299.
doi: 10.1023/A:1008614816374. |
[7] |
R. C. Entringer,
A combinatorial interpretation of the {E}uler and Bernoulli numbers, Nieuw Arch. Wisk. (3), 14 (1966), 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] |
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. |
[12] |
G. Hetyei,
On the $cd$-variation polynomials of André and Simsun permutations, Discrete Comput. Geom., 16 (1996), 259-275.
doi: 10.1007/BF02711512. |
[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. |
[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. |
[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. |
[17] |
C. Poupard,
Two other interpretations of the Entringer numbers, European J. Combin., 18 (1997), 939-943.
doi: 10.1006/eujc.1997.0147. |
[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. |
[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.
|
[21] |
T. A. Springer,
Remarks on a combinatorial problem, Nieuw Arch. Wisk. (3), 19 (1971), 30-36.
|
[22] |
R. P. Stanley,
Flag $f$-vectors and the $cd$-index, Math. Z., 216 (1994), 483-499.
doi: 10.1007/BF02572336. |
[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. |

1 | 2 | 3 | 4 | 5 | 6 | 7 | ||
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 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | ||
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 |
-6 | -5 | -4 | -3 | -2 | -1 | 1 | 2 | 3 | 4 | 5 | 6 | ||
1 | 1 | 1 | |||||||||||
2 | 0 | 1 | 1 | 2 | |||||||||
3 | 0 | 2 | 3 | 3 | 4 | 4 | |||||||
4 | 0 | 4 | 8 | 11 | 11 | 14 | 16 | 16 | |||||
5 | 0 | 16 | 32 | 46 | 57 | 57 | 68 | 76 | 80 | 80 | |||
6 | 0 | 80 | 160 | 236 | 304 | 361 | 361 | 418 | 464 | 496 | 512 | 512 |
-6 | -5 | -4 | -3 | -2 | -1 | 1 | 2 | 3 | 4 | 5 | 6 | ||
1 | 1 | 1 | |||||||||||
2 | 0 | 1 | 1 | 2 | |||||||||
3 | 0 | 2 | 3 | 3 | 4 | 4 | |||||||
4 | 0 | 4 | 8 | 11 | 11 | 14 | 16 | 16 | |||||
5 | 0 | 16 | 32 | 46 | 57 | 57 | 68 | 76 | 80 | 80 | |||
6 | 0 | 80 | 160 | 236 | 304 | 361 | 361 | 418 | 464 | 496 | 512 | 512 |
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