A124293 Number of free generators of degree n of symmetric polynomials in 5-noncommuting variables.

1, 1, 2, 6, 22, 91, 406, 1896, 9093, 44279, 217500, 1073657, 5314870, 26352107, 130778039, 649352929, 3225196431, 16021584848, 79597062632, 395469296912, 1964908443531, 9762920818182, 48508934285620, 241027326818991, 1197601448443963, 5950578465799856

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 <=5

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..500
• 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.
• N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, 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 (10,-32,37,-11).

Crossrefs

Cf. A055105, A055106, A055107, A074664, A001519, A124292, A124294, A124295.

Programs

Magma

I:=[1,1,2,6]; [n le 4 select I[n] else 10*Self(n-1)-32*Self(n-2)+37*Self(n-3)-11*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jan 09 2016

Maple

a:= n-> (Matrix([[6,2,1,1]]). Matrix(4, (i,j)-> if i=j-1 then 1 elif j=1 then [10, -32, 37, -11][i] else 0 fi)^(n-1))[1,4]: seq(a(n), n=1..30); # Alois P. Heinz, Sep 05 2008

Mathematica

LinearRecurrence[{10, -32, 37, -11}, {1, 1, 2, 6}, 30] (* Jean-François Alcover, Jan 08 2016 *)

Formula

O.g.f. (1-9q+24q^2-19q^3)/(1-10q+32q^2-37q^3+11q^4) = (1 - 1/(sum_{k=0}^5 q^k/(prod_{i=1}^k (1-i*q))))/q a(n) = add( A055105(n,k), k=1..5) = add(A055106(n,k),k=1..4)