A262078 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), k>=0, k<=n<=k*(k+1)/2, read by columns.

1, 1, 1, 3, 1, 4, 10, 60, 1, 5, 15, 140, 280, 1260, 12600, 1, 6, 21, 224, 630, 3780, 34650, 110880, 360360, 2522520, 37837800, 1, 7, 28, 336, 1050, 7392, 74844, 276276, 1513512, 9459450, 131171040, 428828400, 2058376320, 9777287520, 97772875200, 2053230379200

Offset

0,4

Links

Alois P. Heinz, Columns k = 0..36, 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), n=k..k*(k+1)/2), k=0..7);

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], {k, 0, 7}, {n, k, k*(k+1)/2}] // Flatten (* Jean-François Alcover, Dec 18 2016, after Alois P. Heinz *)

Crossrefs

Row sums give A007837.

Column sums give A262073.

Cf. A000217, A002024, A262071, A262072 (same read by rows).

Sequence in context: A137405 A322456 A301701 * A121922 A054631 A180063

Adjacent sequences: A262075 A262076 A262077 * A262079 A262080 A262081

Keyword

nonn,tabf

Author

Alois P. Heinz, Sep 10 2015

Status

approved