Abstract
We define a new basis of the algebra of quasi-symmetric functions by lifting the cycle-index polynomials of symmetric groups to noncommutative polynomials with coefficients in the algebra of free quasi-symmetric functions, and then projecting the coefficients to QSym. By duality, we obtain a basis of noncommutative symmetric functions, for which a product formula and a recurrence in the form of a combinatorial complex are obtained. This basis allows to identify noncommutative symmetric functions with the quotient of \(\mathrm{FQSym}\) induced by the pattern-replacement relation \(321\equiv 231\) and \(312\equiv 132\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. Berg, N. Bergeron, F. Saliola, L. Serrano, and M. Zabrocki, A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions, Canad. J. Math. 66 (2014), 525–565.
A. Björner and M. Wachs, \(q\)-hook-length formulas for forests, J. Combinatorial Theory, Ser. A 52 165–187 (1989)
L. Carlitz, Generalized Stirling numbers, Combinatorial Analysis Notes, Duke University, 1968, 1–7.
V. Fazel-Rezai, Equivalence Classes of Permutations Modulo Replacements Between 123 and Two-Integer Patterns.arXiv:1309.4802, 2013.
I. M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. S. Retakh, and J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math. 112, 1995, 218–348.
D. Grinberg, Private communication, 2017.
F. Hivert, J.-C. Novelli, and J.-Y. Thibon, The algebra of binary search trees, Theoretical Computer Science 339 (2005), 129–165.
F. Hivert, J.-C. Novelli, L. Tevlin, and J.-Y. Thibon, Permutation statistics related to a class of noncommutative symmetric functions and generalizations of the Genocchi numbers, Selecta Math. (N.S.) 15 (2009), no. 1, 105–119.
D. Krob, B. Leclerc, and J.-Y. Thibon, Noncommutative symmetric functions II: Transformations of alphabets, Intern. J. Alg. Comput. 7 no. 2, (1997), 181–264.
D. Krob and J.-Y.Thibon, Noncommutative symmetric functions IV : Quantum linear groups and Hecke algebras at \(q=0\), J. Alg. Comb. 6 (1997), 339–376.
W. Kuszmaul, Counting Permutations Modulo Pattern-Replacement Equivalences for Three-Letter Patterns, Electronic Journal of Combinatorics 20 (4) (2013), #P10. arXiv:1304.5667v2.
W. Kuszmaul and Z. Zhou, Equivalence Classes in \(S_n\) for Three Families of Pattern-Replacement Relations, MIT PRIMES, 2013. http://web.mit.edu/primes/materials/2012/Kuszmaul-Zhou.pdf+
A. Lascoux, B. Leclerc, and J.-Y. Thibon, Crystal graphs and \(q\)-analogues of weight multiplicities for the root system \(A_n\), Lett. Math. Phys. 35 (1995), 359–374.
P. Littelmann, A plactic algebra for semisimple Lie algebras, Adv. Math., 124 (1996), 312–331.
S. Linton, J. Propp, T. Roby, and J. West, Equivalence Relations of Permutations Generated by Constrained Transpositions. DMTCS Proceedings, North America, July 2010. http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/view/dmAN0168+
J.-L. Loday, Exponential series without denominators, Lie Theory and its Applications in Physics, IX International Workshop 2013. https://doi.org/10.1007/978-4-431-54270-4_7.
M. Lothaire, Combinatorics on Words, 2nd Ed., Cambridge University press, 1997.
I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Clarendon Press, Oxford 1995.
J.-C. Novelli, J.-Y. Thibon, and F. Toumazet, Noncommutative Bell polynomials and the dual immaculate basis, Algebraic Combinatorics 1 (2018) no. 5, 653–676.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of \(m\)-permutations, \((m+1)\)-ary trees, and \(m\)-parking functions, Advances in Applied Mathematics 117 (2020), 102019.
J. Nzeutchap, Correspondances de Schensted-Fomin, Algèbres de Hopf et graphes gradués en dualité, Thèse de Doctorat, Université de Rouen, 2008.
A. Pierrot, D. Rossin, and J. West, Adjacent transformations in permutations. FPSAC 2011 Proceedings, Discrete Math. Theor. Comput. Sci. Proc., 2011. http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/view/dmAO0167/3638
R. Schimming and S. Z. Rida, Noncommutative Bell polynomials, Internat. J. Algebra Comput. 6 (1996), 635–644.
R. Stanley, An equivalence relation on the symmetric group and multiplicity-free flag \(h\)-vectors, Journal of Combinatorics 3 (2012) no. 3, 277–298. arXiv:1208.3540, 2012.
The On-Line Encyclopedia of Integer Sequences, published electronically at http://oeis.org, 2010.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Novelli, JC., Thibon, JY. & Toumazet, F. A Noncommutative Cycle Index and New Bases of Quasi-symmetric Functions and Noncommutative Symmetric Functions. Ann. Comb. 24, 557–576 (2020). https://doi.org/10.1007/s00026-020-00504-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00026-020-00504-5