A074664 -id:A074664 - cp's OEIS frontend

A087903 Triangle read by rows of the numbers T(n,k) (n > 1, 0 < k < n) of set partitions of n of length k which do not have a proper subset of parts with a union equal to a subset {1,2,...,j} with j < n.

1, 1, 1, 1, 4, 1, 1, 11, 9, 1, 1, 26, 48, 16, 1, 1, 57, 202, 140, 25, 1, 1, 120, 747, 916, 325, 36, 1, 1, 247, 2559, 5071, 3045, 651, 49, 1, 1, 502, 8362, 25300, 23480, 8260, 1176, 64, 1, 1, 1013, 26520, 117962, 159736, 84456, 19404, 1968, 81, 1, 1, 2036, 82509, 525608, 998830, 749154, 253764, 40944, 3105, 100, 1

Offset: 2

Mike Zabrocki, Oct 14 2003

Another version of the triangle T(n,k), 0 <= k <= n, given by [1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, ...] DELTA [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is the operator defined in A084938; see also A086329 for a triangle transposed. - Philippe Deléham, Jun 13 2004

			T(2,1)=1 for {12};
T(3,1)=1, T(3,2) = 1 for {123}; {13|2};
T(4,1)=1, T(4,2)=4, T(4,3)=1 for {1234}; {14|23}, {13|24}, {124|3}, {134|2}; {14|2|3}.
From _Philippe Deléham_, Jul 16 2007: (Start)
Triangle begins:
  1;
  1,    1;
  1,    4,     1;
  1,   11,     9,      1;
  1,   26,    48,     16,      1;
  1,   57,   202,    140,     25,     1;
  1,  120,   747,    916,    325,    36,     1;
  1,  247,  2559,   5071,   3045,   651,    49,    1;
  1,  502,  8362,  25300,  23480,  8260,  1176,   64,  1;
  1, 1013, 26520, 117962, 159736, 84456, 19404, 1968, 81, 1;
  ...
Triangle T(n,k), 0 <= k <= n, given by [1,0,2,0,3,0,4,0,...] DELTA [0,1,0,1,0,1,0,...] begins:
  1;
  1,    0;
  1,    1,     0;
  1,    4,     1,      0;
  1,   11,     9,      1,      0;
  1,   26,    48,     16,      1,     0;
  1,   57,   202,    140,     25,     1,     0;
  1,  120,   747,    916,    325,    36,     1,    0;
  1,  247,  2559,   5071,   3045,   651,    49,    1,  0;
  1,  502,  8362,  25300,  23480,  8260,  1176,   64,  1, 0;
  1, 1013, 26520, 117962, 159736, 84456, 19404, 1968, 81, 1, 0;
  ...
(End)

G. C. Greubel, Rows n = 2..52 of the triangle, flattened
V. E. Adler, Set partitions and integrable hierarchies, arXiv:1510.02900 [nlin.SI], 2015.
Mercedes H. Rosas and Bruce E. Sagan, Symmetric functions in noncommuting variables, arXiv:math/0208168 [math.CO], 2002, 2004.
Yves Le Jan, Loop clusters on complete graphs, arXiv:2504.16976 [math.PR], 2025. See p. 6.
Margarete C. Wolf, Symmetric Functions of Non-commutative Elements, Duke Math. J., 2 (1936), 626-637.

Cf. A000110, A000295, A008277, A055106, A074664.

Cf. A055105.

A := proc(n,k) option remember; local j,ell; if n<=0 or k>=n then 0; elif k=1 or k=n-1 then 1; else S2(n-1,k)+add(add((k-ell-1)*A(n-j-1,k-ell)*S2(j,ell),ell=0..k-1),j=0..n-2); fi; end: S2 := (n,k)->if n<0 or k>n then 0; elif k=n or k=1 then 1 else k*S2(n-1,k)+S2(n-1,k-1); fi:

nmax = 12; t[n_, k_] := t[n, k] = StirlingS2[n-1, k] + Sum[ (k-d-1)*t[n-j-1, k-d]*StirlingS2[j, d], {d, 0, k-1}, {j, 0, n-2}]; Flatten[ Table[ t[n, k], {n, 2, nmax}, {k, 1, n-1}]] (* Jean-François Alcover, Oct 04 2011, after given formula *)

@CachedFunction # T = A087903
def T(n,k): return stirling_number2(n-1, k) + sum( sum( (k-m-1)*T(n-j-1, k-m)*stirling_number2(j, m) for m in (0..k-1) ) for j in (0..n-2) )
flatten([[T(n, k) for k in (1..n-1)] for n in (2..14)]) # G. C. Greubel, Jun 21 2022

T(n, n-1) = T(n,1) = 1.
T(n, n-2) = (n-2)^2.
T(n, 2) = A000295(n).
T(n, k) = S2(n-1, k) + Sum_{j=0..n-2} Sum_{d=0..k-1} (k-d-1)*T(n-j-1, k-d)*S2(j, d), where S2(n, k) is the Stirling number of the second kind.
Sum_{k = 1..n-1} T(n, k) = A074664(n). - Philippe Deléham, Jun 13 2004
G.f.: 1-1/(1+add(add(q^n t^k S2(n, k), k=1..n), n >= 1)) where S2(n, k) are the Stirling numbers of the 2nd kind A008277. - Mike Zabrocki, Sep 03 2005

A124292 Number of free generators of degree n of symmetric polynomials in 4 noncommuting variables.

Original entry on oeis.org

1, 1, 2, 6, 21, 78, 297, 1143, 4419, 17118, 66366, 257391, 998406, 3873015, 15024609, 58285737, 226111986, 877174110, 3402893997, 13201132950, 51212274057, 198672129783, 770725711035, 2989941920334, 11599136512038, 44997518922327, 174562710686622

Offset: 1

Mike Zabrocki, Oct 24 2006

nonn

Also the number of non-splitable set partitions (see Bergeron et al. reference) of length <= 4.
Also the number of nonisomorphic graded posets with 0 and 1 of rank n with no 3-element antichain. - Richard Stanley, Nov 30 2011
Also the number of nonisomorphic graded posets with 0 of rank n+1 with no 3-element antichain. (Using Stanley's definition of graded, that all maximal chains have length n.) - David Nacin, Feb 26 2012

R. Stanley, Enumerative combinatorics. Vol. 1, Cambridge University Press, Cambridge, 1997, pages 96-100.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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; MR2398749, Cand. J. Math 60 (2008) 266-296.
Yibo Gao and Kaarel Hänni, Boolean elements in the Bruhat order, arXiv:2007.08490 [math.CO], 2020.
R. P. Stanley, An Equivalence Relation on the Symmetric Group and Multiplicity-free Flag h-Vectors
M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626-637.
Index entries for linear recurrences with constant coefficients, signature (6,-9,3).

Cf. A055105, A055106, A055107, A074664, A001519, A124293, A124294, A124295.

a:= n-> (Matrix([[2,1,1]]). Matrix(3, (i,j)-> if i=j-1 then 1 elif j=1 then [6,-9,3][i] else 0 fi)^(n-1))[1,3]: seq(a(n), n=1..26); # Alois P. Heinz, Sep 05 2008

m = {{2, 1, 1}, {1, 3, 0}, {1, 1, 1}}; Table[MatrixPower[m, n][[1,1]], {n, 0, 40}] (* David Nacin, Feb 11 2012 *)
LinearRecurrence[{6, -9, 3}, {1, 1, 2}, 70] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2012 *)

def a(n, adict={1:1, 2:1, 3:2}):
    if n in adict:
        return adict[n]
    adict[n]=6*a(n-1)-9*a(n-2)+3*a(n-3)
    return adict[n] # David Nacin, Mar 04 2012

O.g.f.: (1 - 5*q + 5*q^2)/(1 - 6*q + 9*q^2 - 3*q^3) = 1 - 1/(Sum_{k=0..4} q^k/(Product_{i=1..k} (1-i*q))).
a(n) = 6*a(n-1) - 9*a(n-2) + 3*a(n-3). - David Nacin, Feb 11 2012
a(n) = A055105(n,1) + A055105(n,2) + A055105(n,3) + A055105(n,4) = A055106(n,1) + A055106(n,2) + A055106(n,3).
Given matrix A = [[2,1,1],[1,3,0],[1,1,1]], a(n+1) = top left entry in A^n. - David Nacin, Feb 11 2012
a(n) = (1/3)*(x^(n-2) + y^(n-2) + z^(n-2)) for x = (2*cos(Pi/18))^2, y = (2*cos(5*Pi/18))^2, and z = (2*cos(7*Pi/18))^2. - Greg Dresden, Jan 28 2023

A055106 Triangle T(n,k) giving number of symmetric polynomials of degree n in k noncommuting variables, n >=2, 2 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 12, 8, 1, 1, 33, 44, 13, 1, 1, 88, 208, 109, 19, 1, 1, 232, 910, 753, 223, 26, 1, 1, 609, 3809, 4674, 2091, 405, 34, 1, 1, 1596, 15521, 27161, 17220, 4926, 677, 43, 1, 1, 4180, 62185, 151134, 130480, 51702, 10342, 1064, 53, 1

Offset: 2

N. J. A. Sloane, Jun 14 2000

A055105 with first column deleted. - Sean A. Irvine, Mar 12 2022

			T(1,1)=1 from Sum x_1; T(2,2)=1 from Sum x_1 x_2; T(3,2)=1 from Sum x_1 x_2 x_1; T(3,3)=1 from Sum x_1 x_2 x_3; ...
1; 1,1; 1,4,1; 1,12,8,1; 1,33,44,13,1; ...

M. C. Wolf, Symmetric Functions of Non-commutative Elements, Duke Math. J., 2 (1936), 626-637.

Row sums are A074664. Cf. A055105, A055107.

More terms from Sean A. Irvine, Mar 12 2022

A091768 Similar to Bell numbers (A000110).

Original entry on oeis.org

1, 2, 6, 22, 92, 426, 2150, 11708, 68282, 423948, 2788230, 19341952, 141003552, 1076787624, 8589843716, 71404154928, 617151121998, 5535236798058, 51426766394244, 494145546973656, 4903432458931118, 50181840470551778, 529009041574922566

Offset: 0

Jon Perry, Mar 06 2004

nonn

Equals row sums of triangle A163946. - Gary W. Adamson, Aug 06 2009

			The Bell numbers can be generated by;
1
1 2
2 3 5
5 7 10 15
where the Bell numbers are the last entry on each line. This last entry is the first entry on the next line and then the last two entries of the previous column are added, e.g. 7=5+2, 10=7+3, 15=10+5.
This version adds ALL of the entries in the previous column to the new entry.
1
1 2
2 4 6
6 10 16 22
where 10=6+2+1+1, 16=10+2+4, 22=16+6

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Juan S. Auli and Sergi Elizalde, Wilf equivalences between vincular patterns in inversion sequences, arXiv:2003.11533 [math.CO], 2020.
Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.
Zhicong Lin, Sherry H. F. Yan, Vincular patterns in inversion sequences, Applied Mathematics and Computation (2020), Vol. 364, 124672.

Close to A074664

Cf. A000110 (Bell Numbers), A033184, A000108, A163946.

nmax=21; b = ConstantArray[0,nmax]; b[[1]]=1; Do[b[[n+1]] = Binomial[2*n, n]/(n+1) + Sum[b[[k+1]]*Binomial[2*n-k-1, n-k-1]*(k+2)/(n+1),{k,0,n-1}],{n,1,nmax-1}]; b (* Vaclav Kotesovec, Mar 13 2014 *)

v=vector(20); for (i=1,20,v[i]=vector(i)); v[1][1]=1; for (i=2,20, v[i][1]=v[i-1][i-1]; for (j=2,i, v[i][j]=v[i][j-1]+sum(k=j-1,i-1,v[k][j-1]))); for (i=1,20,print1(","v[i][i]))

a(n)=binomial(2*n,n)/(n+1)+sum(k=0,n-1,a(k)*binomial(2*n-k-1,n-k-1)*(k+2)/(n+1)) \\ Paul D. Hanna, Aug 13 2008

a(n)=local(A=1+x*O(x^n),C=serreverse(x-x^2+x^2*O(x^n))/x); for(i=0,n,A=C+x*C^2*subst(A,x,x*C));polcoeff(A,n) \\ Paul D. Hanna, Aug 13 2008

From Paul D. Hanna, Aug 13 2008: (Start)
G.f. satisfies: (1-x)*A(x-x^2) = 1 + x*A(x).
G.f. satisfies: A(x) = C(x) + x*C(x)^2*A(x*C(x)), where C(x) is the Catalan function (A000108).
a(n) = A000108(n) + Sum_{k=0..n-1} a(k)*C(2*n-k-1,n-k-1)*(k+2)/(n+1) for n>=0; eigensequence (shift left) of the Catalan triangle A033184. (End)

More terms from Vincenzo Librandi, Mar 15 2014

A090365 Shifts 1 place left under the INVERT transform of the BINOMIAL transform of this sequence.

Original entry on oeis.org

1, 1, 3, 11, 47, 225, 1177, 6625, 39723, 251939, 1681535, 11764185, 86002177, 655305697, 5193232611, 42726002123, 364338045647, 3215471252769, 29331858429241, 276224445794785, 2682395337435723, 26832698102762435, 276221586866499839, 2923468922184615897

Offset: 0

Paul D. Hanna, Nov 26 2003

nonn

The Hankel transform of this sequence is A000178(n+1); example: det([1,1,3; 1,3,11; 3,11,47]) = 12. - Philippe Deléham, Mar 02 2005
a(n) appears to be the number of indecomposable permutations (A003319) of [n+1] that avoid both of the dashed patterns 32-41 and 41-32. - David Callan, Aug 27 2014
This is true: A nonempty permutation avoids 32-41 and 41-32 if and only if all its components do so. So if A(x) denotes the g.f. for indecomposable {32-41,41-32}-avoiders, then F(x):=1/(1-A(x)) is the g.f. for all {32-41,41-32}-avoiders. From A074664, F(x)=1/x(1-1/B(x)) where B(x) is the o.g.f. for the Bell numbers. Solve for A(x). - David Callan, Jul 21 2017
The Hankel transform of this sequence without the a(0)=1 term is also A000178(n+1). - Michael Somos, Oct 02 2024

Alois P. Heinz, Table of n, a(n) for n = 0..574

Cf. A090366, A090367.
Cf. A074664.
Similar recurrences: A006014, A124758, A217924, A243499, A284005, A290615, A329369, A341392, A347204, A347205.

bintr:= proc(p) proc(n) add(p(k) *binomial(n,k), k=0..n) end end:
invtr:= proc(p) local b;
           b:= proc(n) option remember; local i;
                `if`(n<1, 1, add(b(n-i) *p(i-1), i=1..n+1))
               end;
        end:
b:= invtr(bintr(a)):
a:= n-> `if`(n<0, 0, b(n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 28 2012

a[n_] := Module[{A, B}, A = 1+x; For[k=1, k <= n, k++, B = (A /. x -> x/(1 - x))/(1-x) + O[x]^n // Normal; A = 1 + x*A*B]; SeriesCoefficient[A, {x, 0, n}]]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Oct 23 2016, adapted from PARI *)

{a(n)=local(A); if(n<0,0,A=1+x+x*O(x^n); for(k=1,n,B=subst(A,x, x/(1-x))/(1-x)+x*O(x^n); A=1+x*A*B);polcoeff(A,n,x))}

G.f.: A(x) satisfies A(x) = 1/(1 - A(x/(1-x))*x/(1-x) ).
a(n) = Sum_{k = 0..n} A085838(n, k). - Philippe Deléham, Jun 04 2004
G.f.: 1/x-1-1/(B(x)-1) where B(x) = g.f. for A000110 the Bell numbers. - Vladeta Jovovic, Aug 08 2004
a(n) = Sum_{k=0..n} A094456(n,k). - Philippe Deléham, Nov 07 2007
G.f.: 1/(1-x/(1-2x/(1-x/(1-3x/(1-x/(1-4x/(1-x/(1-5x/(1-... (continued fraction). - Paul Barry, Feb 25 2010
From Sergei N. Gladkovskii, Jan 06 2012  - May 12 2013: (Start)
Continued fractions:
G.f.: 1 - x/(G(0)+x); G(k) = x - 1 + x*k + x*(x-1+x*k)/G(k+1).
G.f.: 1/x - 1/2 + (x^2-4)/(4*U(0)-2*x^2+8) where U(k) = k*(2*k+3)*x^2 + x - 2 - (2-x+2*k*x)*(2+3*x+2*k*x)*(k+1)*x^2/U(k+1).
G.f.: 1/x+1/(U(0)-1) where U(k) = -x*k + 1 - x - x^2*(k+1)/U(k+1).
G.f.: (1 - U(0))/x - 1 where U(k) = 1 - x*(k+2) - x^2*(k+1)/U(k+1).
G.f.: (1 - U(0))/x where U(k) = 1 - x*(k+1)/(1-x/U(k+1)).
G.f.: 1/x + 1/( G(0)-1) where G(k) = 1 - x/(1 - x*(2*k+1)/(1 - x/(1 - x*(2*k+2)/ G(k+1) ))).
G.f.:1/x + 1/( G(0) - 1 ) where G(k) = 1 - x/(1 - x*(k+1)/G(k+1) ).
G.f.: (1 - Q(0))/x where Q(k) = 1 + x/(x*k - 1 )/Q(k+1).
G.f.: 1/x - 1/x/Q(0), where Q(k) = 1 + x/(1 - x + x*(k+1)/(x - 1/Q(k+1))).
(End)
Conjecture: a(n) = b(2^(n-1) - 1) for n > 0 with a(0) = 1 where b(n) = b((n - 2^f(n))/2) + b(floor((2n - 2^f(n))/2)) + b(A025480(n-1)) for n > 0 with b(0) = 1 and where f(n) = A007814(n). - Mikhail Kurkov, Jan 11 2022

A122369 Dimension of 5-variable non-commutative harmonics (twisted derivative). The dimension of the space of non-commutative polynomials in 5 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).

Original entry on oeis.org

1, 4, 19, 93, 459, 2273, 11274, 55964, 277924, 1380527, 6858356, 34074280, 169297743, 841173845, 4179517118, 20766807551, 103184684826, 512698227699, 2547469553647, 12657750705603, 62893284231103, 312501512711984, 1552744642741738, 7715214279423070

Offset: 0

Mike Zabrocki, Aug 30 2006

nonn

			a(1) = 4 because x1-x2, x2-x3, x3-x4, x4-x5 are all of degree 1 and are killed by the differential operator d_x1+d_x2+d_x3+d_x4+d_x5.

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.

N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math.CO/0502082 , Canad. J. Math. 60 (2008), no. 2, 266-296

Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122369, A122370, A122371, A122372.

coeffs(convert(series((1-6*q+11*q^2-6*q^3)/(1-10*q+32*q^2-37*q^3+11*q^4),q,30),`+`)-O(q^30),q);

gf = With[{n = 5}, 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}]]; CoefficientList[gf + O[q]^22, q] (* Jean-François Alcover, Nov 17 2018 *)

G.f. (1-6*q+11*q^2-6*q^3)/(1-10*q+32*q^2-37*q^3+11*q^4) 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=5.

A109062 Triangle read by rows: number of atomic set compositions of size n and length k (see description in A095989) 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 1, 11, 23, 13, 1, 26, 112, 158, 71, 1, 57, 446, 1170, 1241, 461, 1, 120, 1593, 6880, 12871, 10912, 3447, 1, 247, 5337, 35503, 103887, 150413, 106031, 29093, 1, 502, 17190, 168982, 724148, 1589266, 1872286, 1128218, 273343, 1, 1013, 54008

Offset: 1

Mike Zabrocki, Aug 24 2005

Also the number of free generators and primitives of the quasi-symmetric functions in non-commuting variables. - Mike Zabrocki, Aug 06 2006
Triangle given by [1,0,2,0,3,0,4,0,5,...] DELTA [1,2,2,3,3,4,4,5,5,6,6,7,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 01 2007
Apparently, the alternating sums vanish for n > 1. - F. Chapoton, Sep 05 2023

			Atomic set compositions a(1,1)=1: [{1}]; a(2,1)=1, a(2,2)=1: [{12}], [{2},{1}]; a(3,1)=1, a(3,2)=4, a(3,3)=3: [{123}], [{2},{13}], [{3}, {12}], [{23}, {1}], [{13},{2}], [{2},{3},{1}], [{3},{1},{2}], [{3},{2},{1}].
Triangle begins:
  1;
  1,  1;
  1,  4,   3;
  1, 11,  23,  13;
  1, 26, 112, 158, 71;
  ...

N. Bergeron and M. Zabrocki, The Hopf algebras of symmetric functions and quasisymmetric functions in non-commutative variables are free and cofree, arXiv:math/0509265 [math.CO], 2005.
Ming-Jian Ding and Bao-Xuan Zhu, Some results related to Hurwitz stability of combinatorial polynomials, Advances in Applied Mathematics, Volume 152, (2024), 102591. See p. 7.

Row sums are equal to A095989, a(n,n) = A003319, a(n,2) = A000295.

Cf. A095989, A059438, A074664, A087903, A008277, A019538.

f:=(n,k)->coeff(coeff(series(1-1/(1+add(add(q^m*t^i*
    Stirling2(m,i)*i!,i=1..m),m=1..n)),q,n+1),q,n),t,k):
seq(seq(f(n,k), k=1..n), n=1..10);

G.f.: 1-1/(1+Sum_{n>=1} Sum_{k=1..n} q^n*t^k*Stirling2(n,k)*k!).

A122368 Dimension of 4-variable non-commutative harmonics (twisted derivative). The dimension of the space of non-commutative polynomials in 4 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).

Original entry on oeis.org

1, 3, 11, 42, 162, 627, 2430, 9423, 36549, 141777, 549990, 2133594, 8276985, 32109534, 124565121, 483235875, 1874657763, 7272519066, 28212902154, 109448714619, 424593725526, 1647162628047, 6389978382405, 24789187818585

Offset: 1

Mike Zabrocki, Aug 30 2006

nonn

Empirical: a(n) is the sum of the greatest elements over all lexicographically greatest elements in all partitions in the canonical basis of the Temperley-Lieb algebra of order n. - John M. Campbell, Oct 17 2017

			a(1) = 3 because x1-x2, x2-x3, x3-x4 are all of degree 1 and are killed by the differential operator d_x1+d_x2+d_x3+d_x4
For example, the canonical basis of the Temperley-Lieb algebra of order 3 is {{{-3, 1}, {-2, -1}, {2, 3}}, {{-3, 3}, {-2, 2}, {-1, 1}}, {{-3, 3}, {-2, -1}, {1, 2}}, {{-3, -2}, {-1, 1}, {2, 3}}, {{-3, -2}, {-1, 3}, {1, 2}}}, and the lexicographically greatest elements among all partitions in this basis are {2, 3}, {-1, 1}, {1, 2}, {2, 3}, and {1, 2}, with a(3) = 3+1+2+3+2 = 11. - _John M. Campbell_, Oct 17 2017

Michael De Vlieger, Table of n, a(n) for n = 1..1699
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
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.
Hanna Mularczyk, Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations, arXiv:1908.04025 [math.CO], 2019.
M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626-637.
Index entries for linear recurrences with constant coefficients, signature (6,-9,3).

Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122369, A122370, A122371, A122372.

coeffs(convert(series((1-3*q+2*q^2)/(1-6*q+9*q^2-3*q^3),q,30),`+`)-O(q^30),q);

LinearRecurrence[{6, -9, 3}, {1, 3, 11}, 24] (* Jean-François Alcover, Sep 22 2017 *)

O.g.f.: (1-3*q+2*q^2)/(1-6*q+9*q^2-3*q^3) 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=4

A122370 Dimension of 6-variable non-commutative harmonics (twisted derivative). The dimension of the space of non-commutative polynomials in 6 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).

Original entry on oeis.org

1, 5, 29, 172, 1026, 6134, 36712, 219847, 1316963, 7890594, 47282065, 283344410, 1698058817, 10176618298, 60990528210, 365532989831, 2190756912988, 13129979193808, 78692862940748, 471636719623539

Offset: 0

Mike Zabrocki, Aug 30 2006

nonn

			a(1) = 5 because x1-x2, x2-x3, x3-x4, x4-x5, x5-x6 are all of degree 1 and are killed by the differential operator d_x1+d_x2+d_x3+d_x4+d_x5+d_x6.

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.

N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math.CO/0502082 , Canad. J. Math. 60 (2008), no. 2, 266-296
Index entries for linear recurrences with constant coefficients, signature (15,-81,192,-189,53)

Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122368, A122369, A122371, A122372.

coeffs(convert(series((1-10*q+35*q^2-50*q^3+24*q^4)/ (1-15*q+81*q^2 -192*q^3+189*q^4 -53*q^5),q,20), `+`) -O(q^20),q)

LinearRecurrence[{15, -81, 192, -189, 53}, {1, 5, 29, 172, 1026}, 20] (* Jean-François Alcover, Sep 22 2017 *)

o.g.f. (1-10*q+35*q^2-50*q^3+24*q^4) / (1-15*q+81*q^2 -192*q^3+189*q^4 -53*q^5) 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=6.

A122371 Dimension of 7-variable non-commutative harmonics (twisted derivative). The dimension of the space of non-commutative polynomials in 7 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).

Original entry on oeis.org

1, 6, 41, 285, 1989, 13901, 97215, 680079, 4758408, 33297267, 233014444, 1630701426, 11412409945, 79870754268, 558989013403, 3912210491549, 27380636068267, 191631324294463, 1341190961828143, 9386756237545989

Offset: 0

Mike Zabrocki, Aug 30 2006

nonn

			a(1) = 6 because x1-x2, x2-x3, x3-x4, x4-x5, x5-x6, x6-x7 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.

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.

N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math.CO/0502082 , Canad. J. Math. 60 (2008), no. 2, 266-296.
Index entries for linear recurrences with constant coefficients, signature (21,-170,669,-1314,1157,-309).

Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122368, A122369, A122370, A122372.

coeffs(convert(series((1-15*q+ 85*q^2-225*q^3+274*q^4-120*q^5) / (1-21*q+170*q^2-669*q^3+1314*q^4-1157*q^5+309*q^6),q,20),`+`)-O(q^20),q);

LinearRecurrence[{21, -170, 669, -1314, 1157, -309}, {1, 6, 41, 285, 1989, 13901}, 20] (* Jean-François Alcover, Sep 22 2017 *)

G.f.: (1-15*q+ 85*q^2-225*q^3+274*q^4-120*q^5) / (1-21*q+170*q^2-669*q^3 +1314*q^4-1157*q^5 +309*q^6) 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=7.

cp's OEIS Frontend

A087903 Triangle read by rows of the numbers T(n,k) (n > 1, 0 < k < n) of set partitions of n of length k which do not have a proper subset of parts with a union equal to a subset {1,2,...,j} with j < n.

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A124292 Number of free generators of degree n of symmetric polynomials in 4 noncommuting variables.

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A055106 Triangle T(n,k) giving number of symmetric polynomials of degree n in k noncommuting variables, n >=2, 2 <= k <= n.

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A091768 Similar to Bell numbers (A000110).

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A090365 Shifts 1 place left under the INVERT transform of the BINOMIAL transform of this sequence.

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A122369 Dimension of 5-variable non-commutative harmonics (twisted derivative). The dimension of the space of non-commutative polynomials in 5 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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A109062 Triangle read by rows: number of atomic set compositions of size n and length k (see description in A095989) 1 <= k <= n.

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