A262072 Number T(n,k) of partitions of an n-set with distinct block sizes and maximal block size equal to k; triangle T(n,k), n>=0, ceiling((sqrt(1+8*n)-1)/2)<=k<=n, read by rows.

1, 1, 1, 3, 1, 4, 1, 10, 5, 1, 60, 15, 6, 1, 140, 21, 7, 1, 280, 224, 28, 8, 1, 1260, 630, 336, 36, 9, 1, 12600, 3780, 1050, 480, 45, 10, 1, 34650, 7392, 1650, 660, 55, 11, 1, 110880, 74844, 12672, 2475, 880, 66, 12, 1, 360360, 276276, 140712, 20592, 3575, 1144, 78, 13, 1

Offset: 0

Author

Alois P. Heinz, Sep 10 2015

Keywords

• nonn
• tabf

Examples

			Triangle T(n,k) begins:
: 1;
:    1;
:       1;
:       3,  1;
:           4,     1;
:          10,     5,    1;
:          60,    15,    6,    1;
:                140,   21,    7,   1;
:                280,  224,   28,   8,  1;
:               1260,  630,  336,  36,  9,  1;
:              12600, 3780, 1050, 480, 45, 10, 1;
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened

Crossrefs

Row sums give A007837.

Column sums give A262073.

Cf. A002024, A262071, A262078 (same read by columns).

Programs

Maple

b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, binomial(n, i)*b(n-i, i-1))))
    end:
T:= (n, k)-> b(n, k) -`if`(k=0, 0, b(n, k-1)):
seq(seq(T(n,k), k=ceil((sqrt(1+8*n)-1)/2)..n), n=0..14);

Mathematica

b[n_, i_] := b[n, i] = If[i*(i+1)/2n, 0, Binomial[n, i]*b[n-i, i-1]]]]; T[n_, k_] := b[n, k] - If[k == 0, 0, b[n, k-1]]; Table[T[n, k], {n, 0, 14}, {k, Ceiling[(Sqrt[1+8*n]-1)/2], n}] // Flatten (* Jean-François Alcover, Feb 04 2017, translated from Maple *)