

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
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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(ni*j, i1, max(m, i*j)), j=0..n/i))
end:
T:= n> (p> seq(coeff(p, x, i), i=1..n))(b(n$2, 0)):


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}];


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



