A306100
Square array T(n,k) = number of plane partitions of n with parts colored in (at most) k colors; n >= 0, k >= 0; read by antidiagonals.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 10, 6, 0, 1, 4, 21, 34, 13, 0, 1, 5, 36, 102, 122, 24, 0, 1, 6, 55, 228, 525, 378, 48, 0, 1, 7, 78, 430, 1540, 2334, 1242, 86, 0, 1, 8, 105, 726, 3605, 8964, 11100, 3690, 160, 0, 1, 9, 136, 1134, 7278, 25980, 56292, 47496, 11266, 282, 0
Offset: 0
The array starts:
[1 1 1 1 1 1 ...] = A000012
[0 1 2 3 4 5 ...] = A001477
[0 3 10 21 36 55 ...] = A014105
[0 6 34 102 228 430 ...] = A067389
[0 13 122 525 1540 3605 ...]
[0 24 378 2334 8964 25980 ...]
[0 48 1242 11100 56292 203280 ...]
A255970
Number T(n,k) of 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, 2, 2, 0, 3, 8, 6, 0, 5, 24, 42, 24, 0, 7, 60, 198, 264, 120, 0, 11, 144, 780, 1848, 1920, 720, 0, 15, 320, 2778, 10512, 18840, 15840, 5040, 0, 22, 702, 9342, 53184, 146760, 208080, 146160, 40320, 0, 30, 1486, 30186, 250128, 999720, 2129040, 2479680, 1491840, 362880
Offset: 0
T(3,1) = 3: 1a1a1a, 2a1a, 1a.
T(3,2) = 8: 1a1a1b, 1a1b1a, 1b1a1a, 1b1b1a, 1b1a1b, 1a1b1b, 2a1b, 2b1a.
T(3,3) = 6: 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a.
Triangle T(n,k) begins:
1;
0, 1;
0, 2, 2;
0, 3, 8, 6;
0, 5, 24, 42, 24;
0, 7, 60, 198, 264, 120;
0, 11, 144, 780, 1848, 1920, 720;
0, 15, 320, 2778, 10512, 18840, 15840, 5040;
0, 22, 702, 9342, 53184, 146760, 208080, 146160, 40320;
...
-
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*binomial(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* Binomial[k, i], {i, 0, k}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)
A319730
Number T(n,k) of plane 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, 3, 2, 0, 6, 11, 3, 0, 13, 48, 33, 5, 0, 24, 165, 212, 75, 7, 0, 48, 573, 1253, 798, 172, 11, 0, 86, 1759, 6114, 6175, 2284, 326, 15, 0, 160, 5473, 29573, 45040, 25697, 6198, 631, 22, 0, 282, 16051, 131488, 289685, 238516, 86189, 14519, 1102, 30
Offset: 0
Triangle T(n,k) begins:
1;
0, 1;
0, 3, 2;
0, 6, 11, 3;
0, 13, 48, 33, 5;
0, 24, 165, 212, 75, 7;
0, 48, 573, 1253, 798, 172, 11;
0, 86, 1759, 6114, 6175, 2284, 326, 15;
0, 160, 5473, 29573, 45040, 25697, 6198, 631, 22;
0, 282, 16051, 131488, 289685, 238516, 86189, 14519, 1102, 30;
...
A319601
Number of plane partitions of n into parts of sorts {1, 2, ... }.
Original entry on oeis.org
1, 1, 7, 46, 427, 4266, 56424, 772888, 12882984, 226946552, 4546150566, 95857685949, 2267911223257, 55671787777419, 1496799152925925, 42138629446101195, 1273072014409479719, 40078261700215782164, 1346081462070844724125, 46899893904515009262290
Offset: 0
Showing 1-4 of 4 results.
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