A276921 Number A(n,k) of ordered set partitions of [n] with at most k elements per block; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 6, 0, 1, 1, 3, 12, 24, 0, 1, 1, 3, 13, 66, 120, 0, 1, 1, 3, 13, 74, 450, 720, 0, 1, 1, 3, 13, 75, 530, 3690, 5040, 0, 1, 1, 3, 13, 75, 540, 4550, 35280, 40320, 0, 1, 1, 3, 13, 75, 541, 4670, 45570, 385560, 362880, 0

Offset: 0

Author

Alois P. Heinz, Sep 22 2016

Keywords

• nonn
• tabl

Examples

			Square array A(n,k) begins:
  1,    1,     1,     1,     1,     1,     1,     1, ...
  0,    1,     1,     1,     1,     1,     1,     1, ...
  0,    2,     3,     3,     3,     3,     3,     3, ...
  0,    6,    12,    13,    13,    13,    13,    13, ...
  0,   24,    66,    74,    75,    75,    75,    75, ...
  0,  120,   450,   530,   540,   541,   541,   541, ...
  0,  720,  3690,  4550,  4670,  4682,  4683,  4683, ...
  0, 5040, 35280, 45570, 47110, 47278, 47292, 47293, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened
• Daniel Birmajer, Juan B. Gil, David S. Kenepp, and Michael D. Weiner, Restricted generating trees for weak orderings, arXiv:2108.04302 [math.CO], 2021.

Crossrefs

Columns k=0..10 give: A000007, A000142, A080599, A189886, A276924, A276925, A276926, A276927, A276928, A276929, A276930.

Main diagonal gives A000670.

Cf. A276922.

Programs

Maple

A:= proc(n, k) option remember; `if`(n=0, 1, add(
       A(n-i, k)*binomial(n, i), i=1..min(n, k)))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

A[n_, k_] := A[n, k] = If[n==0, 1, Sum[A[n-i, k]*Binomial[n, i], {i, 1, Min[n, k]}]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 03 2017, translated from Maple *)

Formula

E.g.f. of column k: 1/(1-Sum_{i=1..k} x^i/i!).

A(n,k) = Sum_{j=0..k} A276922(n,j).