A055107 -id:A055107 - cp's OEIS frontend

A122372 Dimension of 8-variable non-commutative harmonics (twisted derivative). The dimension of the space of non-commutative polynomials in 8 variables which are killed by all symmetric differential operators (where for a monomial w, d_{xi} ( xi w ) = w and d_{xi} ( xj w ) = 0 for i/=j).

1, 7, 55, 438, 3498, 27962, 223604, 1788406, 14305102, 114429193, 915366442, 7322521512, 58577537621, 468602617723, 3748697751384, 29988696932490, 239903055854075, 1919175464438065, 15353030007717639, 122821355074655309

Offset: 0

Mike Zabrocki, Aug 30 2006

nonn

			A122371 a(1) = 7 because x1-x2, x2-x3, x3-x4, x4-x5, x5-x6, x6-x7, x7-x8 are all of degree 1 and are killed by the differential operator d_x1+d_x2+d_x3+d_x4+d_x5+d_x6+d_x7.

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.
C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778-782.
M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626-637.
Index entries for linear recurrences with constant coefficients, signature (28,-316,1845,-5925,10190,-8249,2119).

Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122368, A122369, A122370, A122371.

coeffs(convert(series((1-21*q+175*q^2-735*q^3+1624*q^4-1764*q^5+720*q^6)/ (1-28*q+316*q^2-1845*q^3+5925*q^4-10190*q^5+8249*q^6-2119*q^7), q,20),`+`)-O(q^20),q);

n = 8; gf = Sum[n!/(n-d)! q^d/Product[(1 - r q), {r, 1, d}], {d, 0, n}]/ Sum[q^d/Product[(1 - r q), {r, 1, d}], {d, 0, n}] + O[q]^20;
CoefficientList[gf, q] (* Jean-François Alcover, Dec 03 2018 *)

G.f.: (1-21*q+175*q^2-735*q^3+1624*q^4-1764*q^5+720*q^6)/ (1-28*q+316*q^2-1845*q^3+5925*q^4-10190*q^5+8249*q^6-2119*q^7) more generally, sum( n!/(n-d)!*q^d/prod((1-r*q),r=1..d), d=0..n) / sum( q^d/prod((1-r*q), r=1..d), d=0..n) where n=8.

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

Original entry on oeis.org

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

Mike Zabrocki, Oct 24 2006

nonn

Also the number of non-splitable set partitions (see Bergeron et al. reference) of length <=5

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).

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

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

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

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

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)

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

Original entry on oeis.org

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

Mike Zabrocki, Oct 24 2006

nonn

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

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).

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

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

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)

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

Original entry on oeis.org

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

Mike Zabrocki, Oct 24 2006

nonn

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

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).

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

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

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).

cp's OEIS Frontend

A122372 Dimension of 8-variable non-commutative harmonics (twisted derivative). The dimension of the space of non-commutative polynomials in 8 variables which are killed by all symmetric differential operators (where for a monomial w, d_{xi} ( xi w ) = w and d_{xi} ( xj w ) = 0 for i/=j).

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

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

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

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