A292745
Number A(n,k) of partitions of n with k sorts of part 1 which are introduced in ascending order; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 2, 1, 1, 3, 6, 5, 2, 1, 1, 3, 7, 13, 7, 4, 1, 1, 3, 7, 19, 26, 11, 4, 1, 1, 3, 7, 20, 52, 54, 15, 7, 1, 1, 3, 7, 20, 62, 151, 108, 22, 8, 1, 1, 3, 7, 20, 63, 217, 442, 219, 30, 12, 1, 1, 3, 7, 20, 63, 232, 803, 1314, 439, 42, 14
Offset: 0
A(3,2) = 6: 3, 21a, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 3, 3, 3, 3, 3, 3, ...
1, 3, 6, 7, 7, 7, 7, 7, 7, ...
2, 5, 13, 19, 20, 20, 20, 20, 20, ...
2, 7, 26, 52, 62, 63, 63, 63, 63, ...
4, 11, 54, 151, 217, 232, 233, 233, 233, ...
4, 15, 108, 442, 803, 944, 965, 966, 966, ...
7, 22, 219, 1314, 3092, 4158, 4425, 4453, 4454, ...
Columns k=0-10 give:
A002865,
A000041,
A320733,
A320734,
A320735,
A320736,
A320737,
A320738,
A320739,
A320740,
A320741.
-
f:= (n, k)-> add(Stirling2(n, j), j=0..k):
b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
f(n, k), add(b(n-i*j, i-1, k), j=0..n/i))
end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);
-
f[n_, k_] := Sum[StirlingS2[n, j], {j, 0, k}];
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i < 2, f[n, k], Sum[b[n - i*j, i - 1, k], {j, 0, n/i}]];
A[n_, k_] := b[n, n, k];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 17 2018, translated from Maple *)
A320817
Number of partitions of n with exactly four sorts of part 1 which are introduced in ascending order.
Original entry on oeis.org
1, 10, 66, 361, 1778, 8207, 36310, 156095, 657785, 2733065, 11241497, 45900679, 186420826, 754165809, 3042167236, 12245294090, 49211278321, 197535872510, 792216674789, 3175088068035, 12719020008668, 50932090504830, 203896407951944, 816089798651203
Offset: 4
-
b:= proc(n, i, k) option remember; `if`(n=0 or i<2, add(
Stirling2(n, j), j=0..k), add(b(n-i*j, i-1, k), j=0..n/i))
end:
a:= n-> (k-> b(n$2, k)-b(n$2, k-1))(4):
seq(a(n), n=4..35);
-
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i < 2, Sum[StirlingS2[n, j], {j, 0, k}], Sum[b[n - i*j, i - 1, k], {j, 0, n/i}]];
a[n_] := With[{k = 4}, b[n, n, k] - b[n, n, k-1]];
a /@ Range[4, 35] (* Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)
A320818
Number of partitions of n with exactly five sorts of part 1 which are introduced in ascending order.
Original entry on oeis.org
1, 15, 141, 1066, 7108, 43747, 255045, 1431320, 7814385, 41804990, 220266447, 1147232914, 5922585396, 30367092789, 154877631181, 786633449995, 3982378528296, 20109428513990, 101339359244739, 509871884291730, 2562078441467318, 12861324297841420
Offset: 5
-
b:= proc(n, i, k) option remember; `if`(n=0 or i<2, add(
Stirling2(n, j), j=0..k), add(b(n-i*j, i-1, k), j=0..n/i))
end:
a:= n-> (k-> b(n$2, k)-b(n$2, k-1))(5):
seq(a(n), n=5..35);
-
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i < 2, Sum[StirlingS2[n, j], {j, 0, k}], Sum[b[n - i*j, i - 1, k], {j, 0, n/i}]];
a[n_] := With[{k = 5}, b[n, n, k] - b[n, n, k - 1]];
a /@ Range[5, 35] (* Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)
Showing 1-3 of 3 results.