OFFSET
0,3
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 linear recurrences with constant coefficients, signature (1,2,-1,-2,-1,2,1,-1).
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
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 *)
PROG
(Sage)
ring = PowerSeriesRing(ZZ, 'x', default_prec=50)
ms = AlternatingGroup(4).molien_series()
list(ring(ms))
# Ralf Stephan, Apr 29 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved