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).

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

Author

Roger Cuculière, Oct 29 2000

Keywords

• nonn
• tabf
• easy
• nice

Examples

			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;
...
		

References

• 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].

Links

• Koda, Tatsuhiko; Sato, Masaki; Takegahara, Yugen; 2-adic properties for the numbers of involutions in the alternating groups, J. Algebra Appl. 14 (2015), no. 4, 1550052 (21 pages).
• Index entries for sequences related to groups

Crossrefs

Cf. A057731 (S_n), A054522, A057741, A051593, A000793.

Column 2, 3, 4 are A048099, A001471, A051695.

See also A061129, A061130, A061131, A061132, A061133, A061134, A061135, A061136, A061137, A061138, A061139, A061140.

Programs

Magma

{* Order(g) : g in Alt(6) *};

Mathematica

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 *)

Extensions

More terms from N. J. A. Sloane, Nov 01 2000

Missing zero in the row for A_9 inserted by N. J. A. Sloane, Mar 27 2015