A265311 a(n) is the number of abelian subgroups of maximal order in S_n.

1, 1, 1, 7, 10, 10, 245, 280, 280, 14700, 15400, 15400, 1401400, 1401400, 1401400, 196196000, 190590400, 190590400, 38022784800, 36212176000, 36212176000, 9759181432000, 9161680528000, 9161680528000, 3206588184800000, 2977546171600000, 2977546171600000

Offset

1,4

Comments

The maximal order of an abelian subgroup in S_n is given by A000792.

Sequence becomes non-monotonic at n=20.

Links

Table of n, a(n) for n=1..27.

Formula

E.g.f.: exp(x^3/3!)*(1 + x^2/2 + 7*x^4/4!) + x - 1.

From Benedict W. J. Irwin, May 24 2016: (Start)

If n=1, a(n) = 1.

If n=2,5,8,11,..., a(n) = n!*2^(-(n+1)/3)*3^((2-n)/3)/Gamma((n+1)/3).

If n=3,6,9,12,..., a(n) = n!*6^(-n/3)/Gamma((n+3)/3).

If n=4,7,10,13,..., a(n) = n!*7*2^(-(n+5)/3)*3^((1-n)/3)/Gamma((n-1)/3).

(End)

Example

a(4) = 7 because we have three cyclic groups: <(1234)> = <(1432)>, <(1243)> = <(1342)>, <(1324)> = <(1423)> and four groups isomorphhic to C_2 X C_2: <(12),(34)>, <(13),(24)>, <(14),(23)> , <(12)(34),(13)(24)> for a total of 7 distinct subgroups of maximal order 4.

Mathematica

nn = 25; Drop[Range[0, nn]! CoefficientList[Series[Exp[x^3/3!] (1 + x^2/2 + 7 x^4/4!) + x - 1, {x, 0, nn}], x], 1]

Crossrefs

Cf. A000792.

Sequence in context: A247191 A317336 A079004 * A192268 A117319 A120645

Adjacent sequences: A265308 A265309 A265310 * A265312 A265313 A265314

Keyword

nonn,easy

Author

Geoffrey Critzer, Dec 06 2015

Status

approved