A124295 Number of free generators of degree n of symmetric polynomials in 7-noncommuting variables.

1, 1, 2, 6, 22, 92, 426, 2145, 11589, 66425, 399682, 2500037, 16115347, 106266473, 712602272, 4837372576, 33128183406, 228308233098, 1580495251012, 10976092266889, 76398165848091, 532614662149795, 3717370694711130

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 <=7

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 (21, -170, 669, -1314, 1157, -309).

Crossrefs

Cf. A055105, A055106, A055107, A074664, A001519, A124292, A124293, A124294.

Programs

Mathematica

LinearRecurrence[{21, -170, 669, -1314, 1157, -309}, {1, 1, 2, 6, 22, 92}, 23] (* Jean-François Alcover, Jan 27 2019 *)

Formula

O.g.f.: (1-20*q+151*q^2-535*q^3+881*q^4-531*q^5) / (1-21*q+170*q^2 -669*q^3 +1314*q^4-1157*q^5+309*q^6) = (1 - 1/(Sum_{k=0..7} q^k/(prod_{i=1}^k (1-i*q))))/q.

a(n) = add( A055105(n,k), k=1..7) = add(A055106(n,k), k=1..6).