A282988 Triangle of partitions of an n-set into boxes of size >= m.

1, 2, 1, 5, 1, 1, 15, 4, 1, 1, 52, 11, 1, 1, 1, 203, 41, 11, 1, 1, 1, 877, 162, 36, 1, 1, 1, 1, 4140, 715, 92, 36, 1, 1, 1, 1, 21147, 3425, 491, 127, 1, 1, 1, 1, 1, 115975, 17722, 2557, 337, 127, 1, 1, 1, 1, 1, 678570, 98253, 11353, 793, 463, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Vladimir Kruchinin, Feb 26 2017

Keywords

• nonn
• tabl

Examples

			Triangle T(n,m) begins:
    1;
    2,   1;
    5,   1,   1;
   15,   4,   1,   1;
   52,  11,   1,   1,   1;
  203,  41,  11,   1,   1,   1;
  877, 162,  36,   1,   1,   1,   1;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

Cf. A000110, A000296, A006505, A057837, A057814, A182931, A260878.

Programs

Maple

T:= proc(n, k) option remember; `if`(n=0, 1, add(
      T(n-j, k)*binomial(n-1, j-1), j=k..n))
    end:
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Sep 28 2017

Mathematica

T[n_, m_] := T[n, m] = Which[Or[n == m, n == 0], 1, m == 0, 0, True, Sum[Binomial[n - 1, i + m - 1] T[n - i - m, m], {i, 0, n - m}]]; Table[T[n, m], {n, 11}, {m, n}] // Flatten (* Michael De Vlieger, Feb 26 2017 *)

Maxima

T(n,m):=if n=m or n=0 then 1 else if m=0 then 0 else sum(binomial(n-1, i+m-1)*T(n-i-m,m), i, 0, n-m);

Formula

T(n,m) = Sum_{i=0..n-m} C(n-1, i+m-1)*T(n-i-m, m).

E.g.f. m column of T(n,m) is exp(exp(x)-Sum_{k=0..m} 1/k!x^k).