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A330759 Number T(n,k) of set partitions into k blocks of strict integer partitions of n; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows. 4
1, 0, 1, 0, 1, 0, 2, 1, 0, 2, 1, 0, 3, 2, 0, 4, 5, 1, 0, 5, 6, 1, 0, 6, 9, 2, 0, 8, 13, 3, 0, 10, 23, 10, 1, 0, 12, 27, 11, 1, 0, 15, 40, 19, 2, 0, 18, 51, 26, 3, 0, 22, 71, 40, 5, 0, 27, 100, 73, 16, 1, 0, 32, 127, 93, 19, 1, 0, 38, 163, 132, 31, 2, 0, 46, 215, 184, 45, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

LINKS

Alois P. Heinz, Rows n = 0..1000, flattened

Wikipedia, Partition of a set

FORMULA

Sum_{k=0..2} T(n,k) = A072706(n).

Sum_{k=1..A003056(n)} k * T(n,k) = A330765(n).

T(A000217(n),n) = 1.

T(A000096(n),n) = A000041(n).

T(n*(n+1)/2+j,n) = A000041(j) for 0 <= j <= n.

EXAMPLE

T(10,1) = 10: (10), 1234, 127, 136, 145, 19, 235, 28, 37, 46.

T(10,2) = 23: 123|4, 124|3, 12|34, 12|7, 134|2, 13|24, 13|6, 14|23, 14|5, 15|4, 16|3, 17|2, 1|234, 1|27, 1|36, 1|45, 1|9, 23|5, 25|3, 2|35, 2|8, 3|7, 4|6.

T(10,3) = 10: 12|3|4, 13|2|4, 14|2|3, 1|23|4, 1|24|3, 1|2|34, 1|2|7, 1|3|6, 1|4|5, 2|3|5.

T(10,4) = 1: 1|2|3|4.

Triangle T(n,k) begins:

  1;

  0,  1;

  0,  1;

  0,  2,   1;

  0,  2,   1;

  0,  3,   2;

  0,  4,   5,  1;

  0,  5,   6,  1;

  0,  6,   9,  2;

  0,  8,  13,  3;

  0, 10,  23, 10,  1;

  0, 12,  27, 11,  1;

  0, 15,  40, 19,  2;

  0, 18,  51, 26,  3;

  0, 22,  71, 40,  5;

  0, 27, 100, 73, 16, 1;

  ...

MAPLE

b:= proc(n, i, k) option remember; `if`(i*(i+1)/2<n, 0,

      `if`(n=0, x^k, b(n, i-1, k) +(t-> b(n-i, t, k)*k

        +b(n-i, t, k+1))(min(n-i, i-1))))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):

seq(T(n), n=0..20);

CROSSREFS

Columns k=0-1 give: A000007, A000009 (for n>0).

Row sums give A294617.

Cf. A000041, A000096, A000217, A003056, A072706, A330460 (another version), A330765.

Sequence in context: A002188 A128313 A283486 * A216607 A025672 A025665

Adjacent sequences:  A330756 A330757 A330758 * A330760 A330761 A330762

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Dec 29 2019

STATUS

approved

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Last modified June 4 00:49 EDT 2020. Contains 334808 sequences. (Running on oeis4.)