Noncommutative Bell polynomials and the dual immaculate basis
Algebraic Combinatorics, Volume 1 (2018) no. 5, pp. 653-676.

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 [], and interpret these in terms of the tridendriform structure of WQSym. We then present a variant of Rey’s self-dual Hopf algebra of set partitions [] adapted to our noncommutative Bell polynomials and give a complete description of the Bell equivalence classes as linear extensions of explicit posets.

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DOI: 10.5802/alco.28
Classification: 16T30, 05E05, 05A18
Keywords: Noncommutative symmetric functions, Quasi-symmetric functions, Bell polynomials, Dendriform algebras
Novelli, Jean-Christophe 1; Thibon, Jean-Yves 1; Toumazet, Frédéric 1

1 Laboratoire d’informatique Gaspard-Monge Université Paris-Est Marne-la-Vallée 5, Boulevard Descartes Champs-sur-Marne 77454 Marne-la-Vallée cedex 2 France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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/articles/10.5802/alco.28/

[1] Baxter, Andrew Michael 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

[2] Berg, Chris; Bergeron, Nantel; Saliola, Franco; Serrano, Luis; Zabrocki, Mike 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 | DOI | MR | Zbl

[3] Berg, Chris; Bergeron, Nantel; Saliola, Franco; Serrano, Luis; Zabrocki, Mike Indecomposable modules for the dual immaculate basis of quasi-symmetric functions, Proc. Am. Math. Soc., Volume 143 (2015) no. 3, pp. 991-1000 | DOI | MR | Zbl

[4] Björner, Anders; Wachs, Michelle L. Permutation statistics and linear extensions of posets, J. Combin. Theory Ser. A, Volume 58 (1991) no. 1, pp. 85-114 | DOI | MR | Zbl

[5] Chouria, Ali; Luque, Jean-Gabriel r-Bell polynomials in combinatorial Hopf algebras, C. R. Math. Acad. Sci. Paris, Volume 355 (2017) no. 3, pp. 243-247 | DOI | MR | Zbl

[6] Claesson, Anders Generalized pattern avoidance, Eur. J. Comb., Volume 22 (2001) no. 7, pp. 961-971 | DOI | MR | Zbl

[7] Désarménien, Jacques Fonctions symétriques associées à des suites classiques de nombres, Ann. Sci. Éc. Norm. Supér., Volume 16 (1983) no. 2, pp. 271-304 | DOI | Numdam | MR | Zbl

[8] Duchamp, Gérard; Hivert, Florent; Thibon, Jean-Yves Noncommutative symmetric functions. VI. Free quasi-symmetric functions and related algebras, Int. J. Algebra Comput., Volume 12 (2002) no. 5, pp. 671-717 | DOI | MR | Zbl

[9] Ebrahimi-Fard, Kurusch; Lundervold, Alexander; Manchon, Dominique Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras, Int. J. Algebra Comput., Volume 24 (2014) no. 5, pp. 671-705 | DOI | MR | Zbl

[10] Foulkes, H. O. Tangent and secant numbers and representations of symmetric groups, Discrete Math., Volume 15 (1976) no. 4, pp. 311-324 | DOI | MR | Zbl

[11] Foulkes, H. O. Eulerian numbers, Newcomb’s problem and representations of symmetric groups, Discrete Math., Volume 30 (1980) no. 1, pp. 3-49 | DOI | MR | Zbl

[12] Gelfand, Israel M.; Krob, Daniel; Lascoux, Alain; Leclerc, Bernard; Retakh, Vladimir S.; Thibon, Jean-Yves Noncommutative symmetric functions, Adv. Math., Volume 112 (1995) no. 2, pp. 218-348 | DOI | MR | Zbl

[13] Gessel, Ira M.; Reutenauer, Christophe Counting permutations with given cycle structure and descent set, J. Combin. Theory Ser. A, Volume 64 (1993) no. 2, pp. 189-215 | DOI | MR | Zbl

[14] Grinberg, Darij Dual creation operators and a dendriform algebra structure on the quasisymmetric functions, Canad. J. Math., Volume 69 (2017) no. 1, pp. 21-53 | DOI | MR | Zbl

[15] Hivert, Florent; Luque, Jean-Gabriel; Novelli, Jean-Christophe; Thibon, Jean-Yves The (1-𝔼)-transform in combinatorial Hopf algebras, J. Algebr. Comb., Volume 33 (2011) no. 2, pp. 277-312 | DOI | MR | Zbl

[16] Hivert, Florent; Novelli, Jean-Christophe; Thibon, Jean-Yves The algebra of binary search trees, Theoret. Comput. Sci., Volume 339 (2005) no. 1, pp. 129-165 | DOI | MR | Zbl

[17] Hivert, Florent; Novelli, Jean-Christophe; Thibon, Jean-Yves Commutative combinatorial Hopf algebras, J. Algebr. Comb., Volume 28 (2008) no. 1, pp. 65-95 | DOI | MR | Zbl

[18] Hivert, Florent; Novelli, Jean-Christophe; Thibon, Jean-Yves Trees, functional equations, and combinatorial Hopf algebras, Eur. J. Comb., Volume 29 (2008) no. 7, pp. 1682-1695 | DOI | MR | Zbl

[19] Josuat-Vergès, Matthieu; Novelli, Jean-Christophe; Thibon, Jean-Yves The algebraic combinatorics of snakes, J. Combin. Theory Ser. A, Volume 119 (2012) no. 8, pp. 1613-1638 | DOI | MR | Zbl

[20] Krob, Daniel; Leclerc, Bernard; Thibon, Jean-Yves Noncommutative symmetric functions. II. Transformations of alphabets, Int. J. Algebra Comput., Volume 7 (1997) no. 2, pp. 181-264 | DOI | MR | Zbl

[21] Krob, Daniel; Thibon, Jean-Yves Noncommutative symmetric functions. IV. Quantum linear groups and Hecke algebras at q=0, J. Algebr. Comb., Volume 6 (1997) no. 4, pp. 339-376 | DOI | MR | Zbl

[22] Loday, Jean-Louis; Ronco, María O. Hopf algebra of the planar binary trees, Adv. Math., Volume 139 (1998) no. 2, pp. 293-309 | DOI | MR | Zbl

[23] Loday, Jean-Louis; Ronco, María O. Order structure on the algebra of permutations and of planar binary trees, J. Algebr. Comb., Volume 15 (2002) no. 3, pp. 253-270 | DOI | MR | Zbl

[24] Macdonald, Ian G. Symmetric functions and Hall polynomials, Oxford Classic Texts in the Physical Sciences, Clarendon Press, 2015, xii+475 pages | MR

[25] Malvenuto, Clauda; Reutenauer, Christophe Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra, Volume 177 (1995) no. 3, pp. 967-982 | DOI | MR | Zbl

[26] Novelli, Jean-Christophe; Thibon, Jean-Yves Construction de trigèbres dendriformes, C. R. Math. Acad. Sci. Paris, Volume 342 (2006) no. 6, pp. 365-369 | DOI | Zbl

[27] Novelli, Jean-Christophe; Thibon, Jean-Yves Polynomial realizations of some trialgebras, 18th Formal Power Series and Algebraic Combinatorics (FPSAC’06) (2006) no. 1, pp. 243-254 | HAL

[28] Novelli, Jean-Christophe; Thibon, Jean-Yves Hopf algebras and dendriform structures arising from parking functions, Fundam. Math., Volume 193 (2007) no. 3, pp. 189-241 | DOI | MR | Zbl

[29] Novelli, Jean-Christophe; Thibon, Jean-Yves Noncommutative symmetric functions and Lagrange inversion, Adv. Appl. Math., Volume 40 (2008) no. 1, pp. 8-35 | DOI | MR | Zbl

[30] Novelli, Jean-Christophe; Thibon, Jean-Yves Superization and (q,t)-specialization in combinatorial Hopf algebras, Electron. J. Comb., Volume 16 (2009) no. 2, Paper no. Research Paper 21, 46 pages http://www.combinatorics.org/volume_16/abstracts/v16i2r21.html | MR | Zbl

[31] Novelli, Jean-Christophe; Thibon, Jean-Yves Binary shuffle bases for quasi-symmetric functions, Ramanujan J., Volume 40 (2016) no. 1, pp. 207-225 | DOI | MR | Zbl

[32] Novelli, Jean-Christophe; Thibon, Jean-Yves Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions (2016) (https://hal.archives-ouvertes.fr/hal-01289784) | Zbl

[33] Nzeutchap, Janvier Correspondances de Schensted-Fomin algébres de Hopf et graphes gradués en dualité, Ph. D. Thesis, Université de Rouen (France) (2008) http://www.theses.fr/2008ROUES046 sous la direction de Florent Hivert, 1 vol. (150 p.)

[34] Rey, Maxime A self-dual Hopf algebra on set partitions (2007) (http://igm.univ-mlv.fr/~rey/articles/hopf_set.pdf)

[35] Rey, Maxime 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] Schimming, Rainer; Rida, Saad Zagloul Noncommutative Bell polynomials, Int. J. Algebra Comput., Volume 6 (1996) no. 5, pp. 635-644 | DOI | MR | Zbl

[37] Sloane, N. J. A. The On-Line Encyclopedia of Integer Sequences, 2010 http://oeis.org | Zbl

[38] Thibon, Jean-Yves Lectures on noncommutative symmetric functions, Interaction of combinatorics and representation theory (MSJ Mem.), Volume 11, Mathematical Society of Japan, 2001, pp. 39-94 | DOI | MR | Zbl

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