A008627 Molien series for A_4.

1, 1, 2, 3, 5, 6, 10, 12, 17, 21, 28, 33, 43, 50, 62, 72, 87, 99, 118, 133, 155, 174, 200, 222, 253, 279, 314, 345, 385, 420, 466, 506, 557, 603, 660, 711, 775, 832, 902, 966, 1043, 1113, 1198, 1275, 1367, 1452, 1552, 1644, 1753, 1853, 1970, 2079, 2205, 2322

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

With offset = 4: a(n) is the number of equivalence classes of compositions (summands >=1) of n into exactly 4 parts where two compositions a,b are considered equivalent if the summands of a can be permuted into the summands of b with an even number of transpositions. For example, let the class representatives be the last such composition in lexicographic order. a(10)=10 because we have the following nine partitions of 10 into 4 parts, {7,1,1,1}, {6,2,1,1}, {5,3,1,1}, {5,2,2,1}, {4,4,1,1}, {4,3,2,1}, {4,2,2,2},{3,3,3,1}, {3,3,2,2} and the class represented by {3,4,2,1}. - Geoffrey Critzer, Oct 16 2012

References

• D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 105.

Links

• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-2,-1,2,1,-1).

Programs

Maple

(1+x^6)/(1-x)/(1-x^2)/(1-x^3)/(1-x^4): seq(coeff(series(%,x,n+1),x,n), n=0..60);

Mathematica

nn=50;CoefficientList[Series[CycleIndex[AlternatingGroup[4],s]/.Table[s[i]->x^i/(1-x^i),{i,1,nn}],{x,0,nn}],x]  (* Geoffrey Critzer, Oct 16 2012 *)

Sage

ring = PowerSeriesRing(ZZ, 'x', default_prec=50)
ms = AlternatingGroup(4).molien_series()
list(ring(ms))
# Ralf Stephan, Apr 29 2014

Formula

a(n) ~ 1/72*n^3. - Ralf Stephan, Apr 29 2014

G.f.: ( 1-x^2+x^4 ) / ( (1+x+x^2)*(1+x)^2*(x-1)^4 ). - R. J. Mathar, Dec 18 2014