A350648
Sum over all partitions of [n] of the number of blocks containing their own index when blocks are ordered with decreasing largest elements.
Original entry on oeis.org
0, 1, 1, 5, 11, 48, 173, 795, 3719, 19343, 106563, 628508, 3923602, 25875858, 179468739, 1305268102, 9925892324, 78728325373, 649856661196, 5571421770478, 49521735963376, 455616186779543, 4332419124871058, 42520560822961111, 430191406640367880
Offset: 0
a(3) = 5 = 3*1 + 2*2: 321, 3|21, 3|2|1; 31|2.
a(4) = 11 = 7*1 + 2*2: 4321, 43|21, 43|2|1, 421|3, 4|321, 4|32|1, 41|3|2; 431|2, 41|32.
-
b:= proc(n, m) option remember; `if`(n=0, [1, 0], add((p->p+
[0, `if`(j=n, p[1], 0)])(b(n-1, max(j, m))), j=1..m+1))
end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..30);
-
b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[Function[p, p + {0, If[j == n, p[[1]], 0]}][b[n - 1, Max[j, m]]], {j, 1, m + 1}]];
a[n_] := b[n, 0][[2]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 11 2022, after Alois P. Heinz *)
A350684
Number T(n,k) of partitions of [n] such that the sum of elements i contained in block i equals k when blocks are ordered with decreasing largest elements; triangle T(n,k), n>=0, 0<=k<=max(0,A008805(n-1)), read by rows.
Original entry on oeis.org
1, 0, 1, 1, 1, 1, 1, 2, 1, 6, 3, 4, 2, 16, 7, 8, 14, 3, 3, 1, 73, 25, 26, 51, 12, 12, 4, 298, 91, 92, 164, 116, 56, 30, 21, 4, 4, 1, 1453, 390, 391, 601, 676, 256, 163, 147, 28, 28, 7, 7366, 1797, 1798, 2484, 3228, 1927, 897, 876, 307, 307, 87, 31, 31, 5, 5, 1
Offset: 0
T(4,0) = 6: 432|1, 42|31, 42|3|1, 4|31|2, 4|3|21, 4|3|2|1.
T(4,1) = 3: 432(1), 42(1)|3, 4(1)|3|2.
T(4,2) = 4: 43|(2)1, 43|(2)|1, 4|3(2)1, 4|3(2)|1,
T(4,3) = 2: 43(1)|(2), 4(1)|3(2).
Triangle T(n,k) begins:
1;
0, 1;
1, 1;
1, 1, 2, 1;
6, 3, 4, 2;
16, 7, 8, 14, 3, 3, 1;
73, 25, 26, 51, 12, 12, 4;
298, 91, 92, 164, 116, 56, 30, 21, 4, 4, 1;
1453, 390, 391, 601, 676, 256, 163, 147, 28, 28, 7;
...
-
b:= proc(n, m) option remember; expand(`if`(n=0, 1, add(
`if`(n=j, x^j, 1)*b(n-1, max(m, j)), j=1..m+1)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..10);
-
b[n_, m_] := b[n, m] = Expand[If[n == 0, 1, Sum[
If[n == j, x^j, 1]*b[n - 1, Max[m, j]], {j, 1, m + 1}]]];
T[n_] := CoefficientList[b[n, 0], x];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 18 2022, after Alois P. Heinz *)
A350649
Number of partitions of [n] avoiding blocks containing their own index when blocks are ordered with decreasing largest elements.
Original entry on oeis.org
1, 0, 1, 1, 6, 16, 73, 298, 1453, 7366, 40689, 238258, 1483306, 9746839, 67415262, 489048716, 3710659737, 29372630485, 242021348787, 2071598497189, 18386889241210, 168944811545046, 1604584556714162, 15731291424746912, 159001720653174800, 1654891767547439393
Offset: 0
a(4) = 6: 432|1, 42|31, 42|3|1, 4|31|2, 4|3|21, 4|3|2|1.
-
b:= proc(n, m) option remember; `if`(n=0, 1, add(
`if`(j=n, 0, b(n-1, max(m, j))), j=1..m+1))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);
-
b[n_, m_] := b[n, m] = If[n == 0, 1, Sum[
If[j == n, 0, b[n-1, Max[m, j]]], {j, 1, m+1}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 16 2022, after Alois P. Heinz *)
A350650
Number of partitions of [n] having exactly one block containing its own index when blocks are ordered with decreasing largest elements.
Original entry on oeis.org
0, 1, 1, 3, 7, 25, 91, 390, 1797, 9069, 49106, 284537, 1751554, 11406588, 78254594, 563642925, 4249337018, 33443545866, 274130245342, 2335311549498, 20637538548167, 188867393030394, 1787189672368355, 17461684290203403, 175930808241047092, 1825666076751872506
Offset: 0
a(4) = 7: 4321, 43|21, 43|2|1, 421|3, 4|321, 4|32|1, 41|3|2.
-
b:= proc(n, m) option remember; series(`if`(n=0, 1, add(
`if`(j=n, x, 1)*b(n-1, max(m, j)), j=1..m+1)), x, 2)
end:
a:= n-> coeff(b(n, 0), x, 1):
seq(a(n), n=0..25);
-
b[n_, m_] := b[n, m] = Series[If[n == 0, 1, Sum[
If[j == n, x, 1]*b[n-1, Max[m, j]], {j, 1, m+1}]], {x, 0, 2}];
a[n_] := Coefficient[b[n, 0], x, 1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 16 2022, after Alois P. Heinz *)
Showing 1-4 of 4 results.