[seqfan] partitions refined by sum of distinct parts

[Richard J. Mathar]

If we construct the number of partitions of n where the
sum of the distinct parts is k we obtain for n>=0 and 0<=k<=n:

  1 
  0   1 
  0   1   1 
  0   1   0   2 
  0   1   1   1   2 
  0   1   0   2   1   3 
  0   1   1   3   1   1   4 
  0   1   0   3   2   2   2   5 
  0   1   1   3   3   2   4   2   6 
  0   1   0   5   2   3   4   4   3   8 
  0   1   1   4   3   4   7   4   5   3  10 
  0   1   0   5   3   4   7   7   6   6   5  12 
  0   1   1   6   4   3  12   6   8   7   9   5  15 
  0   1   0   6   4   5  10  10   9  10  11  10   7  18 

The diagonal is A000009, the sum of the rows A000041, the column k=1 A000012
(partition in all-1's), the column k=2 A000035 (partition where every part=2)
This way of counting appears for example in P. J. Rossky, Martin Karplus,
J. Chem. Phys. 64 (1976) 1569 doi:10.1016/1.432387 equation (16)(1).
Apparently not in the OEIS (?)

# Maple
A := proc(n,k)
    local a,p,pset;
    a := 0 ;
    for p in combinat[partition](n) do
        pset := convert(p,set) ;
        if add(i,i=op(pset)) = k then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:

Richard Mathar