We define a new family of noncommutative Bell polynomials in the algebra of free quasi-symmetric functions and relate it to the dual immaculate basis of quasi-symmetric functions. We obtain noncommutative versions of Grinberg’s results [14], and interpret these in terms of the tridendriform structure of . We then present a variant of Rey’s self-dual Hopf algebra of set partitions [35] adapted to our noncommutative Bell polynomials and give a complete description of the Bell equivalence classes as linear extensions of explicit posets.
Revised: 2018-02-26
Accepted: 2018-06-17
Published online: 2018-11-30
DOI: https://doi.org/10.5802/alco.28
Classification: 16T30, 05E05, 05A18
Keywords: Noncommutative symmetric functions, Quasi-symmetric functions, Bell polynomials, Dendriform algebras
@article{ALCO_2018__1_5_653_0,
author = {Novelli, Jean-Christophe and Thibon, Jean-Yves and Toumazet, Fr\'ed\'eric},
title = {Noncommutative Bell polynomials and the dual immaculate basis},
journal = {Algebraic Combinatorics},
pages = {653--676},
publisher = {MathOA foundation},
volume = {1},
number = {5},
year = {2018},
doi = {10.5802/alco.28},
zbl = {06987761},
mrnumber = {3887406},
language = {en},
url = {https://alco.centre-mersenne.org/item/ALCO_2018__1_5_653_0/}
}
Novelli, Jean-Christophe; Thibon, Jean-Yves; Toumazet, Frédéric. Noncommutative Bell polynomials and the dual immaculate basis. Algebraic Combinatorics, Volume 1 (2018) no. 5, pp. 653-676. doi : 10.5802/alco.28. https://alco.centre-mersenne.org/item/ALCO_2018__1_5_653_0/
[1] Algorithms for permutation statistics, ProQuest LLC, Ann Arbor, MI, 2011, 170 pages (Thesis (Ph.D.)–Rutgers The State University of New Jersey - New Brunswick (USA)) | MR 2942279
[2] A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions, Canad. J. Math., Volume 66 (2014) no. 3, pp. 525-565 | Article | MR 3194160 | Zbl 1291.05206
[3] Indecomposable modules for the dual immaculate basis of quasi-symmetric functions, Proc. Am. Math. Soc., Volume 143 (2015) no. 3, pp. 991-1000 | Article | MR 3293717 | Zbl 1306.05243
[4] Permutation statistics and linear extensions of posets, J. Combin. Theory Ser. A, Volume 58 (1991) no. 1, pp. 85-114 | Article | MR 1119703 | Zbl 0742.05084
[5] -Bell polynomials in combinatorial Hopf algebras, C. R. Math. Acad. Sci. Paris, Volume 355 (2017) no. 3, pp. 243-247 | Article | MR 3621249 | Zbl 1370.16030
[6] Generalized pattern avoidance, Eur. J. Comb., Volume 22 (2001) no. 7, pp. 961-971 | Article | MR 1857258 | Zbl 0994.05004
[7] Fonctions symétriques associées à des suites classiques de nombres, Ann. Sci. Éc. Norm. Supér., Volume 16 (1983) no. 2, pp. 271-304 | Article | Numdam | MR 732346 | Zbl 0525.05006
[8] Noncommutative symmetric functions. VI. Free quasi-symmetric functions and related algebras, Int. J. Algebra Comput., Volume 12 (2002) no. 5, pp. 671-717 | Article | MR 1935570 | Zbl 1027.05107
[9] Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras, Int. J. Algebra Comput., Volume 24 (2014) no. 5, pp. 671-705 | Article | MR 3254718 | Zbl 1309.16023
[10] Tangent and secant numbers and representations of symmetric groups, Discrete Math., Volume 15 (1976) no. 4, pp. 311-324 | Article | MR 0406810 | Zbl 0342.20005
[11] Eulerian numbers, Newcomb’s problem and representations of symmetric groups, Discrete Math., Volume 30 (1980) no. 1, pp. 3-49 | Article | MR 561763 | Zbl 0445.05008
[12] Noncommutative symmetric functions, Adv. Math., Volume 112 (1995) no. 2, pp. 218-348 | Article | MR 1327096 | Zbl 0831.05063
[13] Counting permutations with given cycle structure and descent set, J. Combin. Theory Ser. A, Volume 64 (1993) no. 2, pp. 189-215 | Article | MR 1245159 | Zbl 0793.05004
[14] Dual creation operators and a dendriform algebra structure on the quasisymmetric functions, Canad. J. Math., Volume 69 (2017) no. 1, pp. 21-53 | Article | MR 3589853 | Zbl 1384.05158
[15] The -transform in combinatorial Hopf algebras, J. Algebr. Comb., Volume 33 (2011) no. 2, pp. 277-312 | Article | MR 2765326 | Zbl 1228.05290
[16] The algebra of binary search trees, Theoret. Comput. Sci., Volume 339 (2005) no. 1, pp. 129-165 | Article | MR 2142078 | Zbl 1072.05052
[17] Commutative combinatorial Hopf algebras, J. Algebr. Comb., Volume 28 (2008) no. 1, pp. 65-95 | Article | MR 2420780 | Zbl 1181.16031
[18] Trees, functional equations, and combinatorial Hopf algebras, Eur. J. Comb., Volume 29 (2008) no. 7, pp. 1682-1695 | Article | MR 2431759 | Zbl 1227.05272
[19] The algebraic combinatorics of snakes, J. Combin. Theory Ser. A, Volume 119 (2012) no. 8, pp. 1613-1638 | Article | MR 2946377 | Zbl 1246.05164
[20] Noncommutative symmetric functions. II. Transformations of alphabets, Int. J. Algebra Comput., Volume 7 (1997) no. 2, pp. 181-264 | Article | MR 1433196 | Zbl 0907.05055
[21] Noncommutative symmetric functions. IV. Quantum linear groups and Hecke algebras at , J. Algebr. Comb., Volume 6 (1997) no. 4, pp. 339-376 | Article | MR 1471894 | Zbl 0881.05120
[22] Hopf algebra of the planar binary trees, Adv. Math., Volume 139 (1998) no. 2, pp. 293-309 | Article | MR 1654173 | Zbl 0926.16032
[23] Order structure on the algebra of permutations and of planar binary trees, J. Algebr. Comb., Volume 15 (2002) no. 3, pp. 253-270 | Article | MR 1900627 | Zbl 0998.05013
[24] Symmetric functions and Hall polynomials, Oxford Classic Texts in the Physical Sciences, Clarendon Press, 2015, xii+475 pages | MR 3443860
[25] Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra, Volume 177 (1995) no. 3, pp. 967-982 | Article | MR 1358493 | Zbl 0838.05100
[26] Construction de trigèbres dendriformes, C. R. Math. Acad. Sci. Paris, Volume 342 (2006) no. 6, pp. 365-369 | Article | Zbl 1101.17003
[27] Polynomial realizations of some trialgebras, 18th Formal Power Series and Algebraic Combinatorics (FPSAC’06) (2006) no. 1, pp. 243-254 https://hal-upec-upem.archives-ouvertes.fr/hal-00622743
[28] Hopf algebras and dendriform structures arising from parking functions, Fundam. Math., Volume 193 (2007) no. 3, pp. 189-241 | Article | MR 2289770 | Zbl 1127.16033
[29] Noncommutative symmetric functions and Lagrange inversion, Adv. Appl. Math., Volume 40 (2008) no. 1, pp. 8-35 | Article | MR 2380843 | Zbl 1133.05101
[30] Superization and -specialization in combinatorial Hopf algebras, Electron. J. Comb., Volume 16 (2009) no. 2, Research Paper 21, 46 pages http://www.combinatorics.org/volume_16/abstracts/v16i2r21.html | MR 2576384 | Zbl 1230.05281
[31] Binary shuffle bases for quasi-symmetric functions, Ramanujan J., Volume 40 (2016) no. 1, pp. 207-225 | Article | MR 3486001 | Zbl 1335.05183
[32] Hopf Algebras of -permutations, -ary trees, and -parking functions (2016) (https://hal.archives-ouvertes.fr/hal-01289784) | Zbl 07183699
[33] Correspondances de Schensted-Fomin algébres de Hopf et graphes gradués en dualité (2008) http://www.theses.fr/2008ROUES046 (sous la direction de Florent Hivert, 1 vol. (150 p.)) (Ph. D. Thesis)
[34] A self-dual Hopf algebra on set partitions (2007) (http://igm.univ-mlv.fr/~rey/articles/hopf_set.pdf)
[35] Algebraic constructions on set partitions (2007), 12 pages (19th Formal Power Series and Algebraic Combinatorics (FPSAC’07), https://hal.archives-ouvertes.fr/hal-00622741)
[36] Noncommutative Bell polynomials, Int. J. Algebra Comput., Volume 6 (1996) no. 5, pp. 635-644 | Article | MR 1419136 | Zbl 0899.33006
[37] The On-Line Encyclopedia of Integer Sequences, 2010 http://oeis.org | Zbl 1044.11108
[38] Lectures on noncommutative symmetric functions, Interaction of combinatorics and representation theory (MSJ Mem.) Volume 11, Mathematical Society of Japan, 2001, pp. 39-94 | Article | MR 1862149 | Zbl 0990.05136
