A057740
Irregular triangle read by rows: T(n,k) is the number of elements of alternating group A_n having order k, for n >= 1, 1 <= k <= A051593(n).
Original entry on oeis.org
1, 1, 1, 0, 2, 1, 3, 8, 1, 15, 20, 0, 24, 1, 45, 80, 90, 144, 1, 105, 350, 630, 504, 210, 720, 1, 315, 1232, 3780, 1344, 5040, 5760, 0, 0, 0, 0, 0, 0, 0, 2688, 1, 1323, 5768, 18900, 3024, 37800, 25920, 0, 40320, 9072, 0, 15120, 0, 0, 24192
Offset: 1
Triangle begins:
1;
1;
1, 0, 2;
1, 3, 8;
1, 15, 20, 0, 24;
1, 45, 80, 90, 144;
1, 105, 350, 630, 504, 210, 720;
1, 315, 1232, 3780, 1344, 5040, 5760, 0, 0, 0, 0, 0, 0, 0, 2688;
1, 1323, 5768, 18900, 3024, 37800, 25920, 0, 40320, 9072, 0, 15120, 0, 0, 24192;
...
- J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].
See also
A061129,
A061130,
A061131,
A061132,
A061133,
A061134,
A061135,
A061136,
A061137,
A061138,
A061139,
A061140.
-
{* Order(g) : g in Alt(6) *};
-
row[n_] := (orders = PermutationOrder /@ GroupElements[AlternatingGroup[n] ]; Table[Count[orders, k], {k, 1, Max[orders]}]); Table[row[n], {n, 1, 9}] // Flatten (* Jean-François Alcover, Aug 31 2016 *)
A326241
Number of degree-n even permutations of order dividing 12.
Original entry on oeis.org
1, 1, 1, 3, 12, 36, 216, 1296, 10368, 78912, 634896, 5572656, 51817536, 477672768, 8268884352, 101752505856, 1417554660096, 20985416983296, 344834432195328, 5096129755468032, 70148917686998016
Offset: 0
For n=3 the a(3)=3 solutions are (1), (1, 2, 3), (1, 3, 2) (permutations in cyclic notation).
- J. Riordan, An Introduction to Combinatorial Analysis, John Wiley & Sons, Inc. New York, 1958 (Chap 4, Problem 22).
Cf.
A053502,
A326242,
A000704,
A061130,
A061131,
A061132,
A048099,
A051695,
A061133,
A061134,
A061135,
A326242.
-
E:= (1/2)*exp(x + (1/2)*x^2 + (1/3)*x^3 + (1/4)*x^4 + (1/6)*x^6+(1/12)*x^(12)) + (1/2)*exp(x - (1/2)*x^2 + (1/3)*x^3 - (1/4)*x^4 - (1/6)*x^6-(1/12)*x^(12)):
S:= series(E,x,31):
seq(coeff(S,x,i)*i!,i=0..30);# Robert Israel, Jul 08 2019
-
With[{nn = 22}, CoefficientList[Series[1/2 Exp[x + x^2/2 + x^3/3 + x^4/4 + x^6/6 +x^12/12]+1/2 Exp[x - x^2/2 + x^3/3 - x^4/4 - x^6/6 - x^12/12], {x, 0, nn}], x]*Range[0, nn]!]
A326242
Number of degree-n odd permutations of order dividing 12.
Original entry on oeis.org
0, 0, 1, 3, 12, 60, 360, 2016, 11088, 73872, 602640, 4411440, 81677376, 934435008, 8100473472, 104370819840, 1448725616640, 15823660179456, 247231858514688, 3703908371910912, 66727356304757760, 1124506454958351360, 19305439846610835456
Offset: 0
For n=3 the a(3)=3 solutions are (1, 2), (2, 3), (1, 3) (permutations in cyclic notation).
Cf.
A053502,
A326242,
A000704,
A061130,
A061131,
A061132,
A048099,
A051695,
A061133,
A061134,
A061135,
A326241.
-
E:= (1/2)*exp(x + (1/2)*x^2 + (1/3)*x^3 + (1/4)*x^4 + (1/6)*x^6+(1/12)*x^(12)) - (1/2)*exp(x - (1/2)*x^2 + (1/3)*x^3 - (1/4)*x^4 - (1/6)*x^6-(1/12)*x^(12)):
S:= series(E,x,31):
seq(coeff(S,x,i)*i!,i=0..30); # Robert Israel, Jul 08 2019
-
With[{nn = 22}, CoefficientList[Series[1/2 Exp[x + x^2/2 + x^3/3 + x^4/4 + x^6/6 +x^12/12]-1/2 Exp[x - x^2/2 + x^3/3 - x^4/4 - x^6/6 - x^12/12], {x, 0, nn}], x]*Range[0, nn]!]
Showing 1-3 of 3 results.