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

0,4

Links

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

Example

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;

Maple

b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, 1,
       b(n, i-1) +`if`(i>n, 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)/2<n, 0, If[n==0, 1, b[n, i-1] + If[i>n, 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 *)

Crossrefs

Row sums give A007837.

Column sums give A262073.

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

Sequence in context: A340072 A079546 A014413 * A321743 A341766 A131632

Adjacent sequences: A262069 A262070 A262071 * A262073 A262074 A262075

Keyword

nonn,tabf

Author

Alois P. Heinz, Sep 10 2015

Status

approved