A369704
Number of pairs (p,q) of partitions of n such that the set of parts in q is a subset of the set of parts in p.
Original entry on oeis.org
1, 1, 2, 4, 8, 13, 28, 43, 84, 137, 243, 372, 684, 1010, 1702, 2620, 4256, 6276, 10134, 14740, 23094, 33742, 51139, 73550, 111303, 158140, 233006, 331099, 481324, 674778, 973928, 1353504, 1925734, 2668263, 3748636, 5153887, 7201684, 9820055, 13572468, 18445878
Offset: 0
a(5) = 13: (11111, 11111), (2111, 11111), (2111, 2111), (2111, 221), (221, 11111), (221, 2111), (221, 221), (311, 11111), (311, 311), (32, 32), (41, 11111), (41, 41), (5, 5).
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b:= proc(n, m, i) option remember; `if`(n=0,
`if`(m=0, 1, 0), `if`(i<1, 0, b(n, m, i-1)+add(
add(b(n-i*j, m-i*h, i-1), h=0..m/i), j=1..n/i)))
end:
a:= n-> b(n$3):
seq(a(n), n=0..42);
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b[n_, m_, i_] := b[n, m, i] = If[n == 0, If[m == 0, 1, 0], If[i < 1, 0, b[n, m, i - 1] + Sum[Sum[b[n - i*j, m - i*h, i - 1], {h, 0, m/i}], { j, 1, n/i}]]];
a[n_] := b[n, n, n];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Feb 29 2024, after Alois P. Heinz *)
A369707
Number of pairs (p,q) of distinct partitions of n such that the set of parts in q is a subset of the set of parts in p.
Original entry on oeis.org
0, 0, 0, 1, 3, 6, 17, 28, 62, 107, 201, 316, 607, 909, 1567, 2444, 4025, 5979, 9749, 14250, 22467, 32950, 50137, 72295, 109728, 156182, 230570, 328089, 477606, 670213, 968324, 1346662, 1917385, 2658120, 3736326, 5139004, 7183707, 9798418, 13546453, 18414693
Offset: 0
a(5) = 6: (2111, 11111), (2111, 221), (221, 11111), (221, 2111), (311, 11111), (41, 11111).
a(6) = 17: (21111, 111111), (21111, 2211), (21111, 222), (2211, 111111), (2211, 21111), (2211, 222), (3111, 111111), (321, 111111), (321, 21111), (321, 2211), (321, 222), (321, 3111), (3111, 33), (321, 33), (411, 111111), (42, 222), (51, 111111).
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b:= proc(n, m, i) option remember; `if`(n=0,
`if`(m=0, 1, 0), `if`(i<1, 0, b(n, m, i-1)+add(
add(b(n-i*j, m-i*h, i-1), h=0..m/i), j=1..n/i)))
end:
a:= n-> b(n$3)-combinat[numbpart](n):
seq(a(n), n=0..42);
-
b[n_, m_, i_] := b[n, m, i] = If[n == 0,
If[m == 0, 1, 0], If[i < 1, 0, b[n, m, i - 1] +
Sum[Sum[b[n - i*j, m - i*h, i - 1], {h, 0, m/i}], {j, 1, n/i}]]];
a[n_] := b[n, n, n] - PartitionsP[n];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Feb 29 2024, after Alois P. Heinz *)
Showing 1-2 of 2 results.