A327643
Number of refinement sequences n -> ... -> {1}^n, where in each step one part is replaced by a partition of itself into two smaller parts (in weakly decreasing order).
Original entry on oeis.org
1, 1, 1, 3, 6, 24, 84, 498, 2220, 15108, 92328, 773580, 5636460, 53563476, 471562512, 5270698716, 52117937052, 637276396764, 7317811499736, 100453675122444, 1276319138168796, 19048874583061716, 270233458572751440, 4442429353548965628, 68384217440167826412
Offset: 1
a(1) = 1:
1
a(2) = 1:
2 -> 11
a(3) = 1:
3 -> 21 -> 111
a(4) = 3:
4 -> 31 -> 211 -> 1111
4 -> 22 -> 112 -> 1111
4 -> 22 -> 211 -> 1111
a(5) = 6:
5 -> 41 -> 311 -> 2111 -> 11111
5 -> 41 -> 221 -> 1121 -> 11111
5 -> 41 -> 221 -> 2111 -> 11111
5 -> 32 -> 212 -> 1112 -> 11111
5 -> 32 -> 212 -> 2111 -> 11111
5 -> 32 -> 311 -> 2111 -> 11111
-
b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,
b(n, i-1, k), 0) +b(i$2, k-1)*b(n-i, min(n-i, i), k))
end:
a:= n-> add(b(n$2, i)*(-1)^(n-1-i)*binomial(n-1, i), i=0..n-1):
seq(a(n), n=1..29);
# second Maple program:
a:= proc(n) option remember; `if`(n=1, 1,
add(a(j)*a(n-j)*binomial(n-2, j-1), j=1..n/2))
end:
seq(a(n), n=1..29);
-
a[n_] := a[n] = Sum[Binomial[n-2, j-1] a[j] a[n-j], {j, n/2}]; a[1] = 1;
Array[a, 25] (* Jean-François Alcover, Apr 28 2020 *)
A327702
Number of refinement sequences n -> ... -> {1}^n, where in each step one part that is the rightmost copy of its size is replaced by a partition of itself into smaller parts (in weakly decreasing order).
Original entry on oeis.org
1, 1, 2, 5, 14, 47, 174, 730, 3300, 16361, 85991, 485982, 2877194, 18064663, 118111993, 810388956, 5755059363, 42643884970, 325468477721, 2576976440845, 20960795772211, 176056148076418, 1514733658531058, 13418942409623726, 121442280888373117, 1128425823360525506
Offset: 1
a(4) = 5:
4 -> 1111
4 -> 211 -> 1111
4 -> 31 -> 1111
4 -> 31 -> 211 -> 1111
4 -> 22 -> 211 -> 1111
-
v:= l-> [seq(`if`(i=1 or l[i]>l[i-1], seq(subs(1=[][], sort(
subsop(i=h[], l))), h=({combinat[partition](l[i])[]}
minus{[l[i]]})), [][]), i=1..nops(l))]:
b:= proc(l) option remember; `if`(max(l)<2, 1, add(b(h), h=v(l))) end:
a:= n-> b([n]):
seq(a(n), n=1..26);
A327697
Number of refinement sequences n -> ... -> {1}^n, where in each step every single part of a nonempty selection of parts is replaced by a partition of itself into smaller parts (in weakly decreasing order).
Original entry on oeis.org
1, 1, 2, 7, 22, 122, 598, 4683, 31148, 292008, 2560274, 30122014, 313694962, 4189079688, 53048837390, 826150653479, 11827659365138, 204993767192252, 3371451881544534, 65337695492942258, 1198123466804343518, 25318312971995895392, 516420623159289735874
Offset: 1
a(1) = 1:
1
a(2) = 1:
2 -> 11
a(3) = 2:
3 -> 111
3 -> 21 -> 111
a(4) = 7:
4 -> 1111
4 -> 211 -> 1111
4 -> 31 -> 1111
4 -> 31 -> 211 -> 1111
4 -> 22 -> 1111
4 -> 22 -> 112 -> 1111
4 -> 22 -> 211 -> 1111
A327698
Number of refinement sequences n -> ... -> {1}^n, where in each step exactly one part is replaced by a partition of itself into smaller parts (in weakly decreasing order).
Original entry on oeis.org
1, 1, 2, 6, 17, 74, 300, 1755, 9360, 65510, 442117, 3802889, 30213386, 294892947, 2789021105, 31360525517, 334374848070, 4184958056248, 50606351991305, 704124800141153, 9452367941048830, 143309007303310536, 2124982437997726705, 35389562541842450218
Offset: 1
a(4) = 6:
4 -> 1111
4 -> 211 -> 1111
4 -> 31 -> 1111
4 -> 31 -> 211 -> 1111
4 -> 22 -> 112 -> 1111
4 -> 22 -> 211 -> 1111
Showing 1-4 of 4 results.
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