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

1,3

Comments

Also the number of non-splitable set partitions (see Bergeron et al. reference) of length <=7

Links

Table of n, a(n) for n=1..23.

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).

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).

Mathematica

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

Crossrefs

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

Sequence in context: A014330 A225294 A124294 * A074664 A367442 A091768

Adjacent sequences: A124292 A124293 A124294 * A124296 A124297 A124298

Keyword

nonn

Author

Mike Zabrocki, Oct 24 2006

Status

approved