ALGEBRAIC COMBINATORICS

Noncommutative Bell polynomials and the dual immaculate basis
Algebraic Combinatorics, Volume 1 (2018) no. 5, p. 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 [14], 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 [35] adapted to our noncommutative Bell polynomials and give a complete description of the Bell equivalence classes as linear extensions of explicit posets.
Received : 2017-12-11
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},
     publisher = {MathOA foundation},
     volume = {1},
     number = {5},
     year = {2018},
     pages = {653-676},
     doi = {10.5802/alco.28},
     zbl = {06987761},
     language = {en},
     url = {https://alco.centre-mersenne.org/item/ALCO_2018__1_5_653_0}
}

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] 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 2942279

[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 | Article | MR 3194160 | Zbl 1291.05206

[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 | Article | MR 3293717 | Zbl 1306.05243

[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 | Article | MR 1119703 | Zbl 0742.05084

[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 | Article | MR 3621249 | Zbl 1370.16030

[6] Claesson, Anders Generalized pattern avoidance, Eur. J. Comb., Volume 22 (2001) no. 7, pp. 961-971 | Article | MR 1857258 | Zbl 0994.05004

[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 | Article | Numdam | MR 732346 | Zbl 0525.05006

[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 | Article | MR 1935570 | Zbl 1027.05107

[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 | Article | MR 3254718 | Zbl 1309.16023

[10] Foulkes, H. O. 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] Foulkes, H. O. 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] 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 | Article | MR 1327096 | Zbl 0831.05063

[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 | Article | MR 1245159 | Zbl 0793.05004

[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 | Article | MR 3589853 | Zbl 1384.05158

[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 | Article | MR 2765326 | Zbl 1228.05290

[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 | Article | MR 2142078 | Zbl 1072.05052

[17] Hivert, Florent; Novelli, Jean-Christophe; Thibon, Jean-Yves Commutative combinatorial Hopf algebras, J. Algebr. Comb., Volume 28 (2008) no. 1, pp. 65-95 | Article | MR 2420780 | Zbl 1181.16031

[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 | Article | MR 2431759 | Zbl 1227.05272

[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 | Article | MR 2946377 | Zbl 1246.05164

[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 | Article | MR 1433196 | Zbl 0907.05055

[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 | Article | MR 1471894 | Zbl 0881.05120

[22] Loday, Jean-Louis; Ronco, María O. Hopf algebra of the planar binary trees, Adv. Math., Volume 139 (1998) no. 2, pp. 293-309 | Article | MR 1654173 | Zbl 0926.16032

[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 | Article | MR 1900627 | Zbl 0998.05013

[24] Macdonald, Ian G. Symmetric functions and Hall polynomials, Clarendon Press, Oxford Classic Texts in the Physical Sciences (2015), xii+475 pages | MR 3443860

[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 | Article | MR 1358493 | Zbl 0838.05100

[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 | Article | Zbl 1101.17003

[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 https://hal-upec-upem.archives-ouvertes.fr/hal-00622743

[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 | Article | MR 2289770 | Zbl 1127.16033

[29] Novelli, Jean-Christophe; Thibon, Jean-Yves Noncommutative symmetric functions and Lagrange inversion, Adv. Appl. Math., Volume 40 (2008) no. 1, pp. 8-35 | Article | MR 2380843 | Zbl 1133.05101

[30] Novelli, Jean-Christophe; Thibon, Jean-Yves Superization and (q,t)-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] Novelli, Jean-Christophe; Thibon, Jean-Yves Binary shuffle bases for quasi-symmetric functions, Ramanujan J., Volume 40 (2016) no. 1, pp. 207-225 | Article | MR 3486001 | Zbl 1335.05183

[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 )

[33] Nzeutchap, Janvier Correspondances de Schensted-Fomin algébres de Hopf et graphes gradués en dualité, Université de Rouen (France) (2008) http://www.theses.fr/2008ROUES046 (Ph. D. Thesis)

[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 | Article | MR 1419136 | Zbl 0899.33006

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

[38] Thibon, Jean-Yves Lectures on noncommutative symmetric functions, Interaction of combinatorics and representation theory, Mathematical Society of Japan (MSJ Mem.) Volume 11 (2001), pp. 39-94 | Article | MR 1862149 | Zbl 0990.05136