×

zbMATH — the first resource for mathematics

Noncommutative symmetric functions and Lagrange inversion. (English) Zbl 1133.05101
Summary: We compute the non-commutative Frobenius characteristic of the natural action of the 0-Hecke algebra on parking functions, and obtain as corollaries various forms of the non-commutative Lagrange inversion formula.

MSC:
05E05 Symmetric functions and generalizations
20C08 Hecke algebras and their representations
05A15 Exact enumeration problems, generating functions
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Brouder, C.; Frabetti, A.; Krattenthaler, C., Non-commutative Hopf algebra of formal diffeomorphisms, Adv. math., 200, 479-524, (2006) · Zbl 1133.16025
[2] Duchamp, G.; Hivert, F.; Thibon, J.-Y., Noncommutative symmetric functions VI: free quasi-symmetric functions and related algebras, Internat. J. algebra comput., 12, 671-717, (2002) · Zbl 1027.05107
[3] Gelfand, I.M.; Krob, D.; Lascoux, A.; Leclerc, B.; Retakh, V.S.; Thibon, J.-Y., Noncommutative symmetric functions, Adv. math., 112, 218-348, (1995) · Zbl 0831.05063
[4] Gessel, I., Noncommutative generalization and q-analog of the Lagrange inversion formula, Trans. amer. math. soc., 257, 2, 455-482, (1980) · Zbl 0459.05014
[5] Graham, T.L.; Knuth, D.E.; Patashnik, O., Concrete mathematics, (1989), Addison-Wesley Reading, MA, second ed., 1994 · Zbl 0668.00003
[6] Haiman, M., Conjectures on the quotient ring by diagonal invariants, J. algebraic combin., 3, 17-36, (1994) · Zbl 0803.13010
[7] Hivert, F., Hecke algebras, difference operators, and quasi-symmetric functions, Adv. math., 155, 181-238, (2000) · Zbl 0990.05129
[8] Hivert, F.; Thiéry, N., Mupad-combinat, an open-source package for research in algebraic combinatorics, Sém. lothar. combin., 51, (2004), 70 p. (electronic) · Zbl 1062.05001
[9] F. Hivert, N.M. Thiéry, Representation theories of some towers of algebras related to the symmetric groups and their Hecke algebras, in: Proc. FPSAC’06, San Diego, 2006
[10] Konheim, A.G.; Weiss, B., An occupancy discipline and applications, SIAM J. appl. math., 14, 1266-1274, (1966) · Zbl 0201.50204
[11] Krob, D.; Leclerc, B.; Thibon, J.-Y., Noncommutative symmetric functions II: transformations of alphabets, Internat. J. algebra comput., 7, 181-264, (1997) · Zbl 0907.05055
[12] Krob, D.; Thibon, J.-Y., Noncommutative symmetric functions IV: quantum linear groups and Hecke algebras at \(q = 0\), J. algebraic combin., 6, 339-376, (1997) · Zbl 0881.05120
[13] Kung, J.P.S.; Yan, C., Gončarov polynomials and parking functions, J. combin. theory ser. A, 102, 16-37, (2003) · Zbl 1017.33003
[14] Lascoux, A., Symmetric functions and combinatorial operators on polynomials, CBMS reg. conf. ser. math., vol. 99, (2003), Amer. Math. Soc. Providence, RI, xii+268 pp · Zbl 1039.05066
[15] Lenart, C., Lagrange inversion and Schur functions, J. algebraic combin., 11, 1, 69-78, (2000) · Zbl 0944.05096
[16] Macdonald, I.G., Symmetric functions and Hall polynomials, (1995), Oxford Univ. Press Oxford · Zbl 0487.20007
[17] J.-C. Novelli, J.-Y. Thibon, A Hopf algebra of parking functions, in: Proc. FPSAC/SFCA 2004, Vancouver (electronic) · Zbl 1355.16029
[18] Novelli, J.-C.; Thibon, J.-Y., Hopf algebras and dendriform structures arising from parking functions, Fund. math., 193, 189-241, (2007) · Zbl 1127.16033
[19] Pak, I.; Postnikov, A.; Retakh, V.S., Noncommutative Lagrange theorem and inversion polynomials, preprint, 1995, available at
[20] Raney, G.N., Functional composition patterns and power series reversion, Trans. amer. math. soc., 94, 441-451, (1960) · Zbl 0131.01402
[21] Sloane, N.J.A., The on-line encyclopedia of integer sequences · Zbl 1274.11001
[22] Stanley, R.P., Enumerative combinatorics, vol. 2, (1999), Cambridge Univ. Press Cambridge · Zbl 0928.05001
[23] Stanley, R.P.; Pitman, J., A polytope related to empirical distributions, plane trees, parking functions, and the associahedron, Discrete comput. geom., 27, 603-634, (2002) · Zbl 1012.52019
[24] Yan, C.H., Generalized parking functions, tree inversions and multicolored graphs, Adv. appl. math., 27, 641-670, (2001) · Zbl 0992.05016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.