Stephen P. Humphries - cp's OEIS frontend

User: Stephen P. Humphries

Stephen P. Humphries has authored 14 sequences. Here are the ten most recent ones:

A225108 Number of pairs (x,y) of elements x of the symmetric group S_{n-1} and y of the symmetric group S_{n} that commute. Here the symmetric group S_{n-m} is to be thought of as the subgroup of the symmetric group S_n which stabilizes n-m+1,n-m+2,...n.

1, 2, 8, 42, 288, 2280, 21600, 226800, 2701440, 35199360, 504403200, 7783776000, 130288435200, 2322678758400, 44286571929600, 894449267712000, 19144352747520000, 431093162852352000, 10224590808047616000, 253873324553232384000, 6602896050191400960000

Offset: 1

Stephen P. Humphries, Jun 20 2013

nonn

We have a formula for the number of pairs (x,y) of elements x of the symmetric group S_{n-m} and y of the symmetric group S_{n} that commute.

			When n=2 every element of S_1 commutes with every element of S_2, so we get a(2) = 2. When n=3 the following are the 8 commuting pairs:
[ Id, Id], [ Id, (1, 2)], [ Id, (1, 3, 2)], [ Id, (1, 2, 3)], [ Id, (1, 3)], [ Id, (2, 3)], [ (1, 2), (1, 2)], [ (1, 2), Id ] where Id is the identity element.

Charles R Greathouse IV, Table of n, a(n) for n = 1..442

s:=0;
for k:=0 to n-1 do
    s:=s+Factorial(n-1)*NumberOfPartitions(n-1-k);
end for;

with(combinat):
a:= n-> (n-1)! * add(numbpart(k), k=0..n-1):
seq(a(n), n=1..25);  # Alois P. Heinz, Jun 27 2013

a[n_] := Sum[(n-1)! PartitionsP[n-1-k], {k, 0, n-1}]; Array[a, 25] (* Jean-François Alcover, Jan 17 2016 *)

a(n)=n--!*sum(k=0,n,numbpart(n-k)) \\ Charles R Greathouse IV, Jun 28 2013

a(n) = Sum_{k=0..n-1} (n-1)!*p(n-1-k) where p is the partition function (A000041).

a(n) = A000142(n-1) * A000070(n-1). - Alois P. Heinz, Jun 27 2013

A156231 Sequence gives the Poincaré series [or Poincare series] of an ordinal Hodge algebra, or algebra with straightening law, for a ring that the braid group on four strands acts on. It is Cohen-Macaulay.

Original entry on oeis.org

1, 0, 6, 8, 24, 48, 106, 192, 369, 624, 1080, 1728, 2787, 4248, 6498, 9528, 13962, 19824, 28066, 38760, 53334, 71936, 96618, 127680, 167983, 218040, 281784, 360024, 458037, 577080, 724098, 900936, 1116636, 1373808, 1684038

Offset: 0

Stephen P. Humphries, Feb 06 2009

			For n=2 the dimension of the degree two part is 6.

Stephen P. Humphries, Action of some braid groups on Hodge algebras. Comm. Algebra 26 (1998), no. 4, pages 1233-1242. See Proposition 3.4

Stephen P. Humphries, Action of some braid groups on Hodge algebras Comm. Algebra 26 (1998), no. 4, pages 1233-1242. See Proposition 3.4 on page 1241.

A053090 is a similar Poincaré series [or Poincare series] for a ring on which the three strand braid groups acts.

G.f.: 1-(-4*x^20+8*x^19+6*x^18-12*x^17-11*x^16-2*x^15+25*x^14+10*x^13 -12*x^12) / ((1+x+x^2)^2*(1+x)^5*(1+x^2)*(1-x)^9) -(-14*x^11-15*x^10 +14*x^9+17*x^8+4*x^7-16*x^6-12*x^5+10*x^4+4*x^3-6*x^2) / ((1+x+x^2)^2*(1+x)^5*(1+x^2)*(1-x)^9).

A048274 Let G be the graph with n vertices, the i-th vertex consisting of all multiples of i <= n, where edges indicate that the vertices intersect; a(n) = |n-(number of edges of G)|.

Original entry on oeis.org

1, 1, 1, 0, 0, 3, 3, 5, 6, 9, 9, 15, 15, 18, 21, 24, 24, 30, 30, 36, 39, 42, 42, 51, 52, 55, 57, 63, 63, 75, 75, 79, 82, 85, 88, 99, 99, 102, 105, 114, 114, 126, 126, 132, 138, 141, 141, 153, 154, 160, 163, 169, 169, 178, 181, 190, 193, 196, 196, 217, 217, 220, 226

Offset: 1

Stephen P. Humphries

nonn

			For n=6 the vertices are {1,2,3,4,5,6},{2,4,6},{3,6},{4},{5},{6}. There are 9 = 5+3+1 edges and so a(6) = |6-9| = 3.

A007987 Number of irreducible words of length 2n in the free group with generators x,y such that the total degree of x and the total degree of y both equal zero.

Original entry on oeis.org

1, 0, 8, 40, 312, 2240, 17280, 134568, 1071000, 8627872, 70302888, 577920200, 4786740112, 39899052960, 334391846048, 2815803070920, 23809393390680, 202061204197632, 1720404406215720, 14690717541313128, 125775000062934552

Offset: 0

Stephen P. Humphries

nonn

Also, co-growth function of a certain group given by Humphries 1997 (page 211).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Stephen P Humphries, Cogrowth of groups and the Dedekind-Frobenius group determinant, Mathematical Proc. Camb. Phil. Soc. (1997) vol. 121, pp. 193-217

CoefficientList[Series[(1 - x)*Hypergeometric2F1[1/12, 5/12, 1,
1728*x^4*(x - 1)*(9*x - 1)*(3*x + 1)^2/(81*x^4 - 36*x^3 - 26*x^2 - 4*x + 1)^3]/(81*x^4 - 36*x^3 - 26*x^2 - 4*x + 1)^(1/4), {x, 0,50}], x] (* G. C. Greubel, Mar 07 2017 *)

For n>0, a(n) = A168597(n) - A168597(n-1) = A002426(n)^2 - A002426(n-1)^2.

G.f.: (1-x)*hypergeom([1/12, 5/12],[1],1728*x^4*(x-1)*(9*x-1)*(3*x+1)^2/(81*x^4-36*x^3-26*x^2-4*x+1)^3)/(81*x^4-36*x^3-26*x^2-4*x+1)^(1/4). - Mark van Hoeij, Apr 10 2014

Formula and further terms from Max Alekseyev, Jun 04 2011

A007990 Poincaré series [or Poincare series] of Lie algebra associated with a certain braid group.

Original entry on oeis.org

3, 6, 18, 42, 94, 180, 348, 602, 1047, 1692, 2737, 4194, 6426, 9450, 13863, 19716, 27933, 38616, 53160, 71748, 96396, 127440, 167704, 217740, 281439, 359654, 457617, 576630, 723592, 900396, 1116033, 1373166, 1683327, 2050212, 2488416, 3002934, 3612072

Offset: 2

Stephen P. Humphries

Colin Barker, Table of n, a(n) for n = 2..1000
S. P. Humphries, Braid groups, infinite Lie algebras of Cartan type and rings of invariants, Topology and its Applications, 95 (3) (1999) pp. 173-205.
Index entries for linear recurrences with constant coefficients, signature (2,3,-6,-5,4,10,4,-12,-8,0,8,12,-4,-10,-4,5,6,-3,-2,1).

Vec(x^2*(3 - 3*x^2 + 6*x^3 + 7*x^4 - 8*x^5 - 6*x^6 - 4*x^7 + 13*x^8 + 8*x^9 - 8*x^11 - 14*x^12 + 6*x^13 + 6*x^14 + 6*x^15 - 3*x^16 - 6*x^17 + 3*x^18) / ((1 - x)^9*(1 + x)^5*(1 + x^2)*(1 + x + x^2)^2) + O(x^50)) \\ Colin Barker, Aug 03 2017

The Humphries paper gives a g.f. with denominator (1-x^4)*(1-x^3)^2*(1-x^2)^4*(1-x)^2. - Ralf Stephan, Jun 11 2005

G.f.: x^2*(3 - 3*x^2 + 6*x^3 + 7*x^4 - 8*x^5 - 6*x^6 - 4*x^7 + 13*x^8 + 8*x^9 - 8*x^11 - 14*x^12 + 6*x^13 + 6*x^14 + 6*x^15 - 3*x^16 - 6*x^17 + 3*x^18) / ((1 - x)^9*(1 + x)^5*(1 + x^2)*(1 + x + x^2)^2). - Colin Barker, Aug 02 2017

More terms from Ralf Stephan, Jun 11 2005

A007991 Poincaré series [or Poincare series] of Lie algebra associated with a certain braid group.

Original entry on oeis.org

1, 2, 7, 18, 42, 88, 176, 324, 581, 984, 1631, 2594, 4055, 6140, 9164

Offset: 2

Stephen P. Humphries

nonn

S. P. Humphries, Braid groups, infinite Lie algebras of Cartan type and rings of invariants, Topology and its Applications, 95 (3) (1999) pp. 173-205.

S. P. Humphries, Home page

A007995 Poincaré series [or Poincare series] of Lie algebra associated with a certain braid group.

Original entry on oeis.org

5, 35, 235, 1380, 6711, 26630, 89695, 266305, 716460, 1780978, 4147915, 9144485, 19231895, 38819255, 75566971, 142424625, 260749275, 464984685, 809559075, 1378889190, 2301674475, 3771041860, 6072576655, 9622868805, 15021940010

Offset: 1

Stephen P. Humphries

nonn

The series in the Humphries paper has zeros interleaved.

S. P. Humphries, Home page
S. P. Humphries, Braid groups, infinite Lie algebras of Cartan type and rings of invariants, Topology and its Applications, 95 (3) (1999) pp. 173-205.

a(n) = (1/87178291200) [n^14 + 91*n^13 + 3731*n^12 + 91091*n^11 + 1474473*n^10 + 16429413*n^9 + 126387833*n^8 + 659772113*n^7 + 430175746*n^6 - 19046231204*n^5 - 113904491064*n^4 + 733785439296*n^3 + 1856912273280*n^2 - 12048593356800*n + 13512635136000 ]. - Ralf Stephan, Jun 11 2005

G.f.: (-45*x^15 +520*x^14 -2705*x^13 +8340*x^12 -16875*x^11 +23430*x^10 -22710*x^9 +15480*x^8 -7695*x^7 +3300*x^6 -1586*x^5 +745*x^4 -235*x^3 +40*x^2 -5*x)/(x-1)^15. - Mark van Hoeij, Oct 21 2011

More terms from Ralf Stephan, Jun 11 2005

A007988 Expansion of (x^6-x^5-x^4+2x^2)/((1-x^3)(1-x^2)^2(1-x)).

Original entry on oeis.org

2, 2, 5, 6, 11, 12, 20, 22, 32, 36, 49, 54, 71, 78, 98, 108, 132, 144, 173, 188, 221, 240, 278, 300, 344, 370, 419, 450, 505, 540, 602, 642, 710, 756, 831, 882, 965, 1022, 1112, 1176, 1274, 1344, 1451, 1528, 1643, 1728, 1852, 1944, 2078, 2178

Offset: 2

Stephen P. Humphries

nonn

Poincaré series [or Poincare series] of Lie algebra associated with a certain braid group.

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
S. P. Humphries, Home page
S. P. Humphries, Braid groups, infinite Lie algebras of Cartan type and rings of invariants, Topology and its Applications, 95 (3) (1999) pp. 173-205.
Index entries for linear recurrences with constant coefficients, signature (1, 2, -1, -2, -1, 2, 1, -1).

[Floor((n+1)*(27*(-1)^n+41+16*n+2*n^2)/144): n in [2..60]]; // Vincenzo Librandi, Mar 04 2014

A007988:=n->floor((n+1)*(27*(-1)^n+41+16*n+2*n^2)/144); seq(A007988(n), n=2..100); # Wesley Ivan Hurt, Feb 26 2014

Drop[CoefficientList[Series[(x^6-x^5-x^4+2x^2)/((1-x^3)(1-x^2)^2(1-x)),{x,0,60}],x],2] (* or *) LinearRecurrence[{1,2,-1,-2,-1,2,1,-1},{2,2,5,6,11,12,20,22},60] (* Harvey P. Dale, Apr 04 2013 *)

a(n) = -25/72+A000217(n+1)/12+A000292(n+1)/12+17*(n+1)/144+3*(n+1)*(-1)^n/16-2*A049347(n+2)/9-(-1)^n/8. [R. J. Mathar, Apr 23 2009]
a(2)=2, a(3)=2, a(4)=5, a(5)=6, a(6)=11, a(7)=12, a(8)=20, a(9)=22; for n>9, a(n) = a(n-1)+ 2*a(n-2)-a(n-3)-2*a(n-4)-a(n-5)+2*a(n-6)+a(n-7)-a(n-8). - Harvey P. Dale, Apr 04 2013
a(n) = floor((n+1)*(27*(-1)^n+41+16*n+2*n^2)/144). - Tani Akinari, Jun 26 2013

More terms from Ralf Stephan, Jun 11 2005

A007994 Poincaré series (or Poincare series) of Lie algebra associated with a certain braid group.

Original entry on oeis.org

4, 22, 110, 453, 1482, 4083, 9926, 21959, 45122, 87347, 160922, 284310, 484528, 800206, 1285462, 2014746, 3088824, 4642092, 6851430, 9946827, 14224030, 20059493, 27927926, 38422769, 52279942, 70405249, 93905842, 124126180

Offset: 1

Stephen P. Humphries

The series in the Humphries paper has zeros interleaved.

S. P. Humphries, Braid groups, infinite Lie algebras of Cartan type and rings of invariants, Topology and its Applications, 95 (3) (1999) pp. 173-205.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
S. P. Humphries, Home page.
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

CoefficientList[Series[(8 x^9 - 54 x^8 + 144 x^7 - 178 x^6 + 60 x^5 + 102 x^4 - 137 x^3 + 70 x^2 - 18 x + 4)/(x - 1)^10, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 14 2013 *)

a(n) = (1/362880)*(n^9 + 36*n^8 + 546*n^7 + 4536*n^6 + 19425*n^5 + 37044*n^4 - 592516*n^3 - 41616*n^2 + 7104384*n - 9434880). - Ralf Stephan, Jun 11 2005

G.f.: x*(8*x^9-54*x^8+144*x^7-178*x^6+60*x^5+102*x^4-137*x^3+70*x^2-18*x+4) / (x-1)^10. - Colin Barker, Nov 28 2012

More terms from Ralf Stephan, Jun 11 2005

A008763 Expansion of g.f.: x^4/((1-x)(1-x^2)^2(1-x^3)).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 4, 7, 9, 14, 17, 24, 29, 38, 45, 57, 66, 81, 93, 111, 126, 148, 166, 192, 214, 244, 270, 305, 335, 375, 410, 455, 495, 546, 591, 648, 699, 762, 819, 889, 952, 1029, 1099, 1183, 1260, 1352, 1436, 1536, 1628, 1736, 1836, 1953, 2061, 2187, 2304, 2439

Offset: 0

N. J. A. Sloane, Simon Plouffe, Stephen P. Humphries

Number of 2 X 2 square partitions of n.
1/((1-x^2)*(1-x^4)^2*(1-x^6)) is the Molien series for 4-dimensional representation of a certain group of order 192 [Nebe, Rains, Sloane, Chap. 7].
Number of ways of writing n as n = p+q+r+s so that p >= q, p >= r, q >= s, r >= s with p, q, r, s >= 1. That is, we can partition n as
pq
rs
with p >= q, p >= r, q >= s, r >= s.
The coefficient of s(2n) in s(n,n) * s(n,n) * s(n,n) * s(n,n) is a(n+4), where s(n) is the Schur function corresponding to the trivial representation, s(n,n) is a Schur function corresponding to the two row partition and * represents the inner or Kronecker product of symmetric functions. - Mike Zabrocki, Dec 22 2005
Let F() be the Fibonacci sequence A000045. Let f([x, y, z, w]) = F(x) * F(y) * F(z) * F(w). Let N([x, y, z, w]) = x^2 + y^2 + z^2 + w^2. Let Q(k) = set of all ordered quadruples of integers [x, y, z, w] such that 1 <= x <= y <= z <= w and N([x, y, z, w]) = k. Let P(n) = set of all unordered triples {q1, q2, q3} of elements of some Q(k) such that max(w1, w2, w3) = n and f(q1) + f(q2) = f(q3). Then a(n-1) is the number of elements of P(n). - Michael Somos, Jan 21 2015
Number of partitions of 2n+2 into 4 parts with alternating parity from smallest to largest (or vice versa). - Wesley Ivan Hurt, Jan 19 2021

			a(7) = 4:
41 32 31 22
11 11 21 21
G.f. = x^4 + x^5 + 3*x^6 + 4*x^7 + 7*x^8 + 9*x^9 + 14*x^10 + 17*x^11 + ...
a(5-1) = 1 because P(5) has only one triple {[1,1,1,5], [2,2,2,4], [1,3,3,3]} of elements from Q(28) where f([1,1,1,5]) = 5, f([2,2,2,4]) = 3, f([1,3,3,3]) = 8, and 5 + 3 = 8. - _Michael Somos_, Jan 21 2015
a(6-1) = 1 because P(6) has only one triple {[1,1,2,6], [2,2,3,5], [1,3,4,4]} of elements from Q(42) where f([1,1,2,6]) = 8, f([2,2,3,5]) = 10, f([1,3,4,4]) = 18 and 8 + 10 = 18. - _Michael Somos_, Jan 21 2015
a(7-1) = 3 because P(7) has three triples. The triple {[1,1,1,7], [2,4,4,4], [3,3,3,5]} from Q(52) where f([1,1,1,7]) = 13, f([2,4,4,4]) = 27, f([3,3,3,5]) = 40 and 13 + 27 = 40. The triple {[1,2,2,7], [2,3,3,6], [1,4,4,5]} from Q(58) where f([1,2,2,7]) = 13, f([2,3,3,6]) = 32, f([1,4,4,5]) = 45 and 13 + 32 = 45. The triple {[1,1,3,7], [2,2,4,6], [1,3,5,5]} from Q(60) where f([1,1,3,7]) = 26, f([2,2,4,6]) = 24, f([1,3,5,5]) = 50 and 26 + 24 = 50. - _Michael Somos_, Jan 21 2015

G. E. Andrews, MacMahon's Partition Analysis II: Fundamental Theorems, Annals Combinatorics, 4 (2000), 327-338.
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis VIII: Plane partition diamonds, Advances Applied Math., 27 (2001), 231-242 (Cor. 2.1, n=1).
S. P. Humphries, Braid groups, infinite Lie algebras of Cartan type and rings of invariants, Topology and its Applications, 95 (3) (1999) pp. 173-205.

T. D. Noe, Table of n, a(n) for n = 0..1000
Nesrine Benyahia-Tani, Zahra Yahi, and Sadek Bouroubi, Ordered and non-ordered non-congruent convex quadrilaterals inscribed in a regular n-gon, Rostocker Math. Kolloq. 68, 71-79 (2013), Theorem 5.
W. Duke, On codes and Siegel modular forms, Int. Math. Res. Notes 1993, No. 5, Theorem 2. [MR1219862 (94d:11029)]
Michele Graffeo, Sergej Monavari, Riccardo Moschetti, and Andrea T. Ricolfi, Enumeration of partitions via socle reduction, arXiv:2501.10267 [math.CO], 2025. See p. 40.
S. P. Humphries, Home page
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 450
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 232
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Michael Somos, In the Elliptic Realm
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-2,-1,2,1,-1).

See A266769 for a version without the four leading zeros.
Cf. A001993, A070557, A070558, A070559, A089299, A001970, A089292, A026810, A001400.
First differences of A097701.
Cf. A082424, A082437.

a:=[0,0,0,0,1,1,3,4];; for n in [9..60] do a[n]:=a[n-1]+2*a[n-2]-a[n-3]-2*a[n-4]-a[n-5]+2*a[n-6]+a[n-7]-a[n-8]; od; a; # G. C. Greubel, Sep 10 2019

K:=Rationals(); M:=MatrixAlgebra(K,4); q1:=DiagonalMatrix(M,[1,-1,1,-1]); p1:=DiagonalMatrix(M,[1,1,-1,-1]); q2:=DiagonalMatrix(M,[1,1,1,-1]); h:=M![1,1,1,1, 1,1,-1,-1, 1,-1,1,-1, 1,-1,-1,1]/2; H:=MatrixGroup<4,K|q1,q2,h,p1>; MolienSeries(H);

R:=PowerSeriesRing(Integers(), 60); [0,0,0,0] cat Coefficients(R!( x^4/((1-x)*(1-x^2)^2*(1-x^3)) )); // G. C. Greubel, Sep 10 2019

a:= n-> (Matrix(8, (i,j)-> if (i=j-1) then 1 elif j=1 then [1,2,-1,-2,-1,2,1,-1][i] else 0 fi)^n)[1,5]: seq(a(n), n=0..60); # Alois P. Heinz, Jul 31 2008

CoefficientList[Series[x^4/((1-x)*(1-x^2)^2*(1-x^3)), {x,0,60}], x] (* Jean-François Alcover, Mar 30 2011 *)
LinearRecurrence[{1,2,-1,-2,-1,2,1,-1},{0,0,0,0,1,1,3,4},60] (* Harvey P. Dale, Mar 04 2012 *)
a[ n_]:= Quotient[9(n+1)(-1)^n +2n^3 -9n +65, 144]; (* Michael Somos, Jan 21 2015 *)
a[ n_]:= Sign[n] SeriesCoefficient[ x^4/((1-x)(1-x^2)^2(1-x^3)), {x, 0, Abs@n}]; (* Michael Somos, Jan 21 2015 *)

{a(n) = (9*(n+1)*(-1)^n + 2*n^3 - 9*n + 65) \ 144}; /* Michael Somos, Jan 21 2015 */

a(n)=([0,1,0,0,0,0,0,0; 0,0,1,0,0,0,0,0; 0,0,0,1,0,0,0,0; 0,0,0,0,1,0,0,0; 0,0,0,0,0,1,0,0; 0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,1; -1,1,2,-1,-2,-1,2,1]^n*[0;0;0;0;1;1;3;4])[1,1] \\ Charles R Greathouse IV, Feb 06 2017

def AA008763_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x^4/((1-x)*(1-x^2)^2*(1-x^3))).list()
AA008763_list(60) # G. C. Greubel, Sep 10 2019

Let f4(n) = number of partitions n = p+q+r+s into exactly 4 parts, with p >= q >= r >= s >= 1 (see A026810, A001400) and let g4(n) be the number with q > r (so that g4(n) = f4(n-2)). Then a(n) = f4(n) + g4(n).
a(n) = (1/144)*( 2*n^3 + 9*n*((-1)^n - 1) - 16*((n is 2 mod 3) - (n is 1 mod 3)) ).
a(n) = (1/72)*(n+3)*(n+2)*(n+1)-(1/12)*(n+2)*(n+1)+(5/144)*(n+1)+(1/16)*(n+1)*(-1)^n+(1/16)*(-1)^(n+1)+(7/144)+(2*sqrt(3)/27)*sin(2*Pi*n/3). - Richard Choulet, Nov 27 2008
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - 2*a(n-4) - a(n-5) + 2*a(n-6) + a(n-7) - a(n-8), n>7. - Harvey P. Dale, Mar 04 2012
a(n) = floor((9*(n+1)*(-1)^n + 2*n^3 - 9*n + 65)/144). - Tani Akinari, Nov 06 2012
a(n+1) - a(n) = A008731(n-3). - R. J. Mathar, Aug 06 2013
a(n) = -a(-n) for all n in Z. - Michael Somos, Jan 21 2015
Euler transform of length 3 sequence [1, 2, 1]. - Michael Somos, Jun 26 2017

Entry revised Dec 25 2003

cp's OEIS Frontend

User: Stephen P. Humphries

A225108 Number of pairs (x,y) of elements x of the symmetric group S_{n-1} and y of the symmetric group S_{n} that commute. Here the symmetric group S_{n-m} is to be thought of as the subgroup of the symmetric group S_n which stabilizes n-m+1,n-m+2,...n.

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A156231 Sequence gives the Poincaré series [or Poincare series] of an ordinal Hodge algebra, or algebra with straightening law, for a ring that the braid group on four strands acts on. It is Cohen-Macaulay.

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A048274 Let G be the graph with n vertices, the i-th vertex consisting of all multiples of i <= n, where edges indicate that the vertices intersect; a(n) = |n-(number of edges of G)|.

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A007987 Number of irreducible words of length 2n in the free group with generators x,y such that the total degree of x and the total degree of y both equal zero.

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A007990 Poincaré series [or Poincare series] of Lie algebra associated with a certain braid group.

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PARI

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A007991 Poincaré series [or Poincare series] of Lie algebra associated with a certain braid group.

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A007995 Poincaré series [or Poincare series] of Lie algebra associated with a certain braid group.

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A007988 Expansion of (x^6-x^5-x^4+2x^2)/((1-x^3)(1-x^2)^2(1-x)).

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Magma

Maple

Mathematica

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A007994 Poincaré series (or Poincare series) of Lie algebra associated with a certain braid group.

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A008763 Expansion of g.f.: x^4/((1-x)(1-x^2)^2(1-x^3)).