A124294 Number of free generators of degree n of symmetric polynomials in 6-noncommuting variables.

1, 1, 2, 6, 22, 92, 425, 2119, 11184, 61499, 347980, 2007643, 11734604, 69181578, 410179429, 2441025998, 14562284120, 87012222100, 520458020949, 3115224471290, 18654716694895, 111741999352603, 669466118302169

Offset: 1

Author

Mike Zabrocki, Oct 24 2006

Keywords

• nonn

Comments

Also the number of non-splitable set partitions (see Bergeron et al. reference) of length <=6

Links

• N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math/0502082 [math.CO], 2005; Canad. J. Math. 60 (2008), no. 2, 266-296.
• M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626-637.
• Index entries for linear recurrences with constant coefficients, signature (15, -81, 192, -189, 53).

Crossrefs

Cf. A055105, A055106, A055107, A074664, A001519, A124292, A124293, A124295.

Programs

Mathematica

LinearRecurrence[{15, -81, 192, -189, 53}, {1, 1, 2, 6, 22}, 23] (* Jean-François Alcover, Dec 04 2018 *)

Formula

O.g.f.: (1-14*q+68*q^2-135*q^3+91*q^4)/(1-15*q+81*q^2-192*q^3+189*q^4-53*q^5) = (1 - 1/(sum_{k=0}^6 q^k/(prod_{i=1}^k (1-i*q))))/q a(n) = add( A055105(n,k), k=1..6) = add(A055106(n,k),k=1..5)