a(n,m)  tabf head (staircase) for  A103921
  
  Number of distinct parts of the partitions of n in Abramowitz-Stegun order

  
  a(n,m)  tabf head (staircase) for  A103921
 
  n\m   1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
 
  1     1  

  2     1  1  

  3     1  2  1  

  4     1  2  1  2  1  

  5     1  2  2  2  2  2  1

  6     1  2  2  1  2  3  1  2  2  2  1  

  7     1  2  2  2  2  3  2  2  2  3  2  2  2  2  1  

  8     1  2  2  2  1  2  3  3  2  2  2  3  2  3  1  2  3  2  2  2  2  1  

  9     1  2  2  2  2  2  3  3  2  2  3  1  2  3  3  3  3  2  2  3  2  3  2  2  3  2  2  2  2  1

 10     1  2  2  2  2  1  2  3  3  3  2  3  2  2  2  3  3  2  3  4  2  2  2  2  3  3  3  3  3  1  2  3  2  3  2  2  3  2  2  2  2  1

   .
   .
   .

  n\m   1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42



 The sequence of row lengths is A000041: [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...] (partition numbers).

 One could add the row for n=0 with a 1, if the part 0 is considered for n=0, and only for this n.

 For the ordering of this tabf array a(n,m) see Abramowitz-Stegun ref. pp. 831-2. 
 E.g. a(4,4) refers to the fourth partition of n=4 in this ordering, namely (1^2,3)=[1,1,3], whence a(4,4)=2 because as a set this list 
 reads {1,2},  which is a 2-element set.