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 *)
A350683
Total sum over all partitions of [n] of elements i contained in block i when blocks are ordered with decreasing largest elements.
Original entry on oeis.org
0, 1, 1, 8, 17, 98, 362, 1916, 9512, 53858, 315872, 1984979, 13105685, 91128546, 663815424, 5055622309, 40148341135, 331753228115, 2846786927873, 25323311882074, 233137061978065, 2218141402504254, 21780561656373552, 220451321425101091, 2297330116404668422
Offset: 0
a(4) = 17 = 3*1 + 4*2 + 2*3: 432(1), 42(1)|3, 4(1)|3|2, 43|(2)1, 43|(2)|1, 4|3(2)1, 4|3(2)|1, 43(1)|(2), 4(1)|3(2).
-
b:= proc(n, m) option remember; `if`(n=0, [1, 0], add((p-> [0,
`if`(n=j, p[1]*j, 0)]+p)(b(n-1, max(m, j))), j=1..m+1))
end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..25);
-
b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[With[{p = b[n-1, Max[m, j]]},
{0, If[n == j, p[[1]]*j, 0]} + p], {j, 1, m+1}]];
a[n_] := b[n, 0][[2]];
Table[a[n], {n, 0, 25}]; (* Jean-François Alcover, May 08 2022, after Alois P. Heinz *)
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