A387082 - OEIS

A387082

Triangle T(n,k) read by rows: T(n,k) is the number of partitions of [n] whose set of block sizes equals [k]; n >= 0, 0 <= k <= A003056(n).

1, 0, 1, 0, 1, 0, 1, 3, 0, 1, 6, 0, 1, 25, 0, 1, 60, 60, 0, 1, 231, 210, 0, 1, 658, 1400, 0, 1, 2619, 7560, 0, 1, 8550, 40320, 12600, 0, 1, 35695, 224070, 69300, 0, 1, 129756, 1315380, 693000, 0, 1, 568503, 7610460, 5405400, 0, 1, 2255344, 46766720, 46666620

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OFFSET

0,8

LINKS

Alois P. Heinz, Rows n = 0..560, flattened

Wikipedia, Partition of a set

FORMULA

T(A000217(n),n) = A022915(n).

EXAMPLE

T(5,2) = 25: 12|34|5, 12|35|4, 12|3|45, 12|3|4|5, 13|24|5, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 15|23|4, 1|23|45, 1|23|4|5, 14|25|3, 14|2|35, 14|2|3|5, 15|24|3, 1|24|35, 1|24|3|5, 15|2|34, 1|25|34, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45.

Triangle T(n,k) begins:

0, 1;

0, 1, 3;

0, 1, 6;

0, 1, 25;

0, 1, 60, 60;

0, 1, 231, 210;

0, 1, 658, 1400;

0, 1, 2619, 7560;

0, 1, 8550, 40320, 12600;

0, 1, 35695, 224070, 69300;

0, 1, 129756, 1315380, 693000;

...

MAPLE

f:= proc(n) option remember; floor((sqrt(1+8*n)-1)/2) end:

T:= proc(n, i) option remember; `if`(k=0, 1-signum(n), `if`(n=0,

`if`(i=0, 1, 0), add(T(n-i*j, i-1)*combinat[multinomial]

(n, n-i*j, i$j)/j!, j=`if`(i=1, n, 1..n/i))))

end:

seq(seq(T(n, k), k=0..f(n)), n=0..14);

CROSSREFS

Row sums give A393098.

Last terms of rows give A394281.

Cf. A000217, A003056, A022915.

Sequence in context: A355172 A102765 A355257 * A393036 A129684 A247255

Adjacent sequences: A387079 A387080 A387081 * A387083 A387084 A387085

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Apr 17 2026

STATUS

approved