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A133800
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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).
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3
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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
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OFFSET
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1,5
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LINKS
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FORMULA
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Take triangle of Stirling numbers of second kind (A008277) and multiply k-th column by A001710(k) (order of alternating group A_k).
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EXAMPLE
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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)
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MAPLE
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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
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MATHEMATICA
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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];
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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