A133800 Triangle read by rows in which row n gives number of ways to partition n labeled elements into k pie slices allowing the pie to be turned over (n >= 1, 1 <= k <= n).

1, 1, 1, 1, 3, 1, 1, 7, 6, 3, 1, 15, 25, 30, 12, 1, 31, 90, 195, 180, 60, 1, 63, 301, 1050, 1680, 1260, 360, 1, 127, 966, 5103, 12600, 15960, 10080, 2520, 1, 255, 3025, 23310, 83412, 158760, 166320, 90720, 20160, 1, 511, 9330, 102315, 510300, 1369620

Offset: 1

Author

Barry Cipra and N. J. A. Sloane, Jan 17 2008

Keywords

• nonn
• tabl
• easy

Examples

			Triangle begins:
1,
1,  1,
1,  3,   1,
1,  7,   6,    3,
1, 15,  25,   30,   12,
1, 31,  90,  195,  180,   60,
1, 63, 301, 1050, 1680, 1260, 360.
...
For row n = 4 we have the following "pies":
. 1
./ \
2 . 3 . 12 .. 12 . 123
.\ / .. / \ .(..)..(..)
. 4 .. 3--4 . 34 .. 4 .. (1234)
k=4 .. k=3 ..k=2 . k=2 . k=1
(3)....(6)...(3)..(4)... (1)
		

Links

• C. G. Bower, Transforms (2)

Crossrefs

Row sums give A032262. Diagonals give A000225, A000392, A032263, A133799, A001710.

Programs

Maple

A001710 := proc(n) if n < 2 then 1; else n!/2 ; fi ; end: A008277 := proc(n,k) combinat[stirling2](n,k) ; end: A133800 := proc(n,k) A008277(n,k)*A001710(k-1) ; end: for n from 1 to 10 do for k from 1 to n do printf("%d, ",A133800(n,k)) ; od: od: # R. J. Mathar, Jan 18 2008

Mathematica

A001710[n_] := If[n<2, 1, n!/2]; A008277[n_, k_] := StirlingS2[n, k]; A133800[n_, k_] := A008277[n, k]*A001710[k-1]; Table[A133800[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 20 2014, after R. J. Mathar *)
(* A (n >= 0, k >= 0)-based version: *)
A133800[n_, k_] := k! StirlingS2[n+1, k+1] / If[k>1, 2, 1];
Table[A133800[n,k], {n,0,9}, {k,0,n}] // Flatten (* Peter Luschny, Oct 19 2017 *)

Formula

Take triangle of Stirling numbers of second kind (A008277) and multiply k-th column by A001710(k) (order of alternating group A_k).

Extensions

More terms from R. J. Mathar, Jan 18 2008