A156042 A(n,k) for n >= k in triangular ordering, where A(n,k) is the number of compositions (ordered partitions) of n into k parts, with the first part greater than or equal to all other parts.

1, 1, 2, 1, 2, 4, 1, 3, 6, 11, 1, 3, 8, 17, 32, 1, 4, 11, 26, 54, 102, 1, 4, 13, 35, 82, 172, 331, 1, 5, 17, 48, 120, 272, 567, 1101, 1, 5, 20, 63, 170, 412, 918, 1906, 3724, 1, 6, 24, 81, 235, 607, 1434, 3152, 6518, 12782, 1, 6, 28, 102, 317, 872, 2180, 5049, 10978, 22616, 44444

Offset: 1

Author

Jack W Grahl, Feb 02 2009

Keywords

• nonn
• tabl

Comments

The value is smaller than the number of compositions (ordered partitions) of n into k parts and at least the number of (unordered) partitions.

Examples

			A(5,3) = 8 and the 8 compositions of 5 into 3 parts with first part maximal are:
[5,0,0], [4,1,0], [4,0,1], [3,2,0], [3,0,2], [3,1,1], [2,2,1], [2,1,2].
1
1  2
1  2  4
1  3  6  11
1  3  8  17  32
1  4  11 26  54  102
		

Links

• Alois P. Heinz, Rows n = 1..100, flattened

Crossrefs

A156041 is the whole of the square. A156043 is the diagonal. See also A156039 and A156040.

Programs

Maple

b:= proc(n,i,m) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i=1 then `if`(n<=m, 1, 0)
    else add(b(n-k, i-1, m), k=0..m)
      fi
    end:
A:= (n,k)-> add(b(n-m, k-1, m), m=ceil(n/k)..n):
seq(seq(A(n,k), k=1..n), n=1..12); # Alois P. Heinz, Jun 14 2009

Mathematica

nn=10; Table[Table[Coefficient[Series[Sum[x^i((1-x^(i+1))/(1-x))^(k-1), {i, 0, n}], {x, 0, nn}], x^n], {k, 1, n}], {n, 1, nn}]//Grid (* Geoffrey Critzer, Jul 15 2013 *)

Extensions

More terms from Alois P. Heinz, Jun 14 2009