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A134979 Triangle read by rows: T(n,k) = number of partitions of n where the maximum number of objects in partitions of any given size is k. 2
1, 0, 2, 0, 1, 2, 0, 1, 1, 3, 0, 0, 3, 2, 2, 0, 0, 2, 4, 1, 4, 0, 0, 1, 6, 3, 3, 2, 0, 0, 1, 6, 4, 6, 1, 4, 0, 0, 0, 6, 7, 8, 3, 3, 3, 0, 0, 0, 5, 7, 14, 4, 6, 2, 4, 0, 0, 0, 5, 7, 18, 7, 9, 5, 3, 2, 0, 0, 0, 3, 10, 22, 9, 14, 6, 6, 1, 6, 0, 0, 0, 2, 9, 26, 15, 19, 11, 9, 3, 5, 2 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Every column is eventually 0; the last row with a nonzero value in column k is A024916(k). T(A024916(k)-i, k) <= P(i), where P is the partition function (A000041); equality holds for 0 <= i <= k. The partition represented by the last number in column k is row k of A010766.
LINKS
EXAMPLE
For the partition [3,2^2], there are 3 objects in the part of size 3 and 4 objects in the parts of size 2, so this partition is counted towards T(7,4).
Triangle T(n,k) begins:
1;
0, 2;
0, 1, 2;
0, 1, 1, 3;
0, 0, 3, 2, 2;
0, 0, 2, 4, 1, 4;
0, 0, 1, 6, 3, 3, 2;
0, 0, 1, 6, 4, 6, 1, 4;
0, 0, 0, 6, 7, 8, 3, 3, 3;
0, 0, 0, 5, 7, 14, 4, 6, 2, 4;
0, 0, 0, 5, 7, 18, 7, 9, 5, 3, 2;
0, 0, 0, 3, 10, 22, 9, 14, 6, 6, 1, 6;
...
MAPLE
b:= proc(n, i, m) option remember; `if`(n=0 or i=1, x^
max(m, n), add(b(n-i*j, i-1, max(m, i*j)), j=0..n/i))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2, 0)):
seq(T(n), n=1..20); # Alois P. Heinz, Feb 07 2020
MATHEMATICA
b[n_, i_, m_] := b[n, i, m] = If[n == 0 || i == 1, x^Max[m, n], Sum[b[n - i j, i - 1, Max[m, i j]], {j, 0, n/i}]];
T[n_] := Table[Coefficient[b[n, n, 0], x, i], {i, 1, n}];
Array[T, 20] // Flatten (* Jean-François Alcover, Nov 10 2020, after Alois P. Heinz *)
CROSSREFS
Cf. A008284, A091602, A000041 (row sums), A000005 (main diagonal), A032741 (2nd diagonal), A010766.
Column sums give A332233.
Sequence in context: A227698 A331466 A166124 * A112248 A244860 A308009
KEYWORD
nonn,look,tabl
AUTHOR
STATUS
approved

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Last modified May 30 15:53 EDT 2024. Contains 372968 sequences. (Running on oeis4.)