Bergeron, Nantel; Zabrocki, Mike The Hopf algebras of symmetric functions and quasi-symmetric functions in non-commutative variables are free and co-free. (English) Zbl 1188.16030 J. Algebra Appl. 8, No. 4, 581-600 (2009). It is a classical result that the algebra of symmetric polynomials in commuting variables is freely generated by the elementary symmetric polynomials. Much work has been done on getting analogous results for the \(Q\)-algebra NCSym of symmetric functions in non-commutative variables, in particular to show that it is freely generated by various monomial bases. Recently, the authors, together with M. Rosas and C. Reutenauer, gave a Hopf algebra structure on NCSym [Can. J. Math. 60, No. 2, 266-296 (2008; Zbl 1180.16025)], and this was used by F. Hivert, J.-C. Novelli and J.-Y. Thibon to give a proof that NCSym is both free and co-free (i.e., the graded dual of NCSym is free) [J. Algebr. Comb. 28, No. 1, 65-95 (2008; Zbl 1181.16031)]. The paper under review shows that NCSym is free and co-free by exhibiting bases which generate the algebra and coalgebra. This involves set partitions and Lyndon words. The generating bases have simple and natural rules for the product of two elements. The authors also study the algebra NCQSym of quasi-symmetric functions in non-commutative variables. This is ordered by set compositions (ordered set partitions). It is also a Hopf algebra, and analogous results are obtained in showing that it is both free and co-free. Again the product of two basis elements has a simple and elegant expression. Reviewer: Earl J. Taft (New Brunswick) Cited in 1 ReviewCited in 34 Documents MSC: 16T30 Connections of Hopf algebras with combinatorics 05E05 Symmetric functions and generalizations 16T05 Hopf algebras and their applications Keywords:noncommutative symmetric functions; noncommutative quasi-symmetric functions; Hopf algebras; set partitions; set compositions; monomial bases Citations:Zbl 1180.16025; Zbl 1181.16031 PDFBibTeX XMLCite \textit{N. Bergeron} and \textit{M. Zabrocki}, J. Algebra Appl. 8, No. 4, 581--600 (2009; Zbl 1188.16030) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: Triangle read by rows: number of atomic set compositions of size n and length k (see description in A095989) 1 <= k <= n. Triangle read by rows of numbers b_{n,k}, n>=1, 1<=k<=n such that Product_{n,k} 1/(1-q^n t^k)^{b_{n,k}} = 1 + Sum_{i,j>=1} S_{i,j} q^i t^j where S_{i,j} are entries in the table A008277 (the inverse Euler transformation of the table of Stirling numbers of the second kind). References: [1] Aguiar M., Fields Institute Monographs 23, in: Coxeter Groups and Hopf Algebras (2006) [2] Bergeron N., Electron. J. Combin. 13 [3] DOI: 10.1016/j.jalgebra.2006.01.032 · Zbl 1113.06001 · doi:10.1016/j.jalgebra.2006.01.032 [4] DOI: 10.4153/CJM-2008-013-4 · Zbl 1180.16025 · doi:10.4153/CJM-2008-013-4 [5] DOI: 10.1007/s10801-007-0077-0 · Zbl 1181.16031 · doi:10.1007/s10801-007-0077-0 [6] DOI: 10.1016/j.crma.2006.01.009 · Zbl 1101.17003 · doi:10.1016/j.crma.2006.01.009 [7] Poirier S., Ann. Sci. Math. Québec 19 pp 7990– [8] Reutenauer C., London Mathematical Society Monographs, New Series 7, in: Free Lie Algebras (1993) [9] DOI: 10.1090/S0002-9947-04-03623-2 · Zbl 1071.05073 · doi:10.1090/S0002-9947-04-03623-2 [10] DOI: 10.1017/CBO9780511609589 · doi:10.1017/CBO9780511609589 [11] DOI: 10.1016/0021-8693(76)90183-6 · doi:10.1016/0021-8693(76)90183-6 [12] DOI: 10.1215/S0012-7094-36-00253-3 · Zbl 0016.00501 · doi:10.1215/S0012-7094-36-00253-3 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.