A256130
Number T(n,k) of partitions of n into parts of exactly k sorts which are introduced in ascending order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 5, 12, 7, 1, 0, 7, 30, 33, 11, 1, 0, 11, 72, 130, 77, 16, 1, 0, 15, 160, 463, 438, 157, 22, 1, 0, 22, 351, 1557, 2216, 1223, 289, 29, 1, 0, 30, 743, 5031, 10422, 8331, 2957, 492, 37, 1, 0, 42, 1561, 15877, 46731, 52078, 26073, 6401, 788, 46, 1
Offset: 0
T(3,1) = 3: 1a1a1a, 2a1a, 3a.
T(3,2) = 4: 1a1a1b, 1a1b1a, 1a1b1b, 2a1b.
T(3,3) = 1: 1a1b1c.
Triangle T(n,k) begins:
1;
0, 1;
0, 2, 1;
0, 3, 4, 1;
0, 5, 12, 7, 1;
0, 7, 30, 33, 11, 1;
0, 11, 72, 130, 77, 16, 1;
0, 15, 160, 463, 438, 157, 22, 1;
0, 22, 351, 1557, 2216, 1223, 289, 29, 1;
0, 30, 743, 5031, 10422, 8331, 2957, 492, 37, 1;
0, 42, 1561, 15877, 46731, 52078, 26073, 6401, 788, 46, 1;
...
Columns k=0-10 give:
A000007,
A000041 (for n>0),
A258457,
A258458,
A258459,
A258460,
A258461,
A258462,
A258463,
A258464,
A258465.
-
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);
-
b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]]; T[n_, k_] := Sum[b[n, n, k-i]*(-1)^i/(i!*(k-i)!), {i, 0, k}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 21 2016, after Alois P. Heinz *)
A319600
Number T(n,k) of plane partitions of n into parts of exactly k sorts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 3, 4, 0, 6, 22, 18, 0, 13, 96, 198, 120, 0, 24, 330, 1272, 1800, 840, 0, 48, 1146, 7518, 19152, 20640, 7920, 0, 86, 3518, 36684, 148200, 274080, 234720, 75600, 0, 160, 10946, 177438, 1080960, 3083640, 4462560, 3180240, 887040, 0, 282, 32102, 788928, 6952440, 28621920, 62056080, 73175760, 44432640, 10886400
Offset: 0
Triangle T(n,k) begins:
1;
0, 1;
0, 3, 4;
0, 6, 22, 18;
0, 13, 96, 198, 120;
0, 24, 330, 1272, 1800, 840;
0, 48, 1146, 7518, 19152, 20640, 7920;
0, 86, 3518, 36684, 148200, 274080, 234720, 75600;
0, 160, 10946, 177438, 1080960, 3083640, 4462560, 3180240, 887040;
...
A319731
Number of plane partitions of n into parts of sorts {1, 2, ... } which are introduced in ascending order.
Original entry on oeis.org
1, 1, 5, 20, 99, 483, 2855, 16759, 112794, 777862, 5864191, 45388575, 381557427, 3265488790, 29815712658, 279926300139, 2762328453142, 27952237049003, 296275051753578, 3212312177119572, 36258222471852860, 419025393587012853, 5010022284030897550
Offset: 0
A319732
Number of plane partitions of 2n into parts of exactly n sorts which are introduced in ascending order.
Original entry on oeis.org
1, 3, 48, 1253, 45040, 2074266, 115308621, 7403931515, 542578637369, 44353623326199, 3992458392860603, 392255543503496555, 41726405940340028501, 4768006168373548992878, 582709500037368041005243, 75765509130126834789261446, 10436240655486571146294062847
Offset: 0
Showing 1-4 of 4 results.
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