A373242 T(n,k) is the sum for all integer partitions of n of length k of the difference between the number of different parts and the number of different multiplicities.

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 1, 0, 0, 0, 3, 2, 1, 0, 0, 0, 0, 3, 4, 3, 1, 0, 0, 0, 0, 4, 6, 4, 2, 2, 0, 0, 0, 0, 4, 8, 8, 5, 1, 1, 0, 0, 0, 0, 5, 10, 10, 7, 2, 1, 1, 0, 0, 0, 0, 5, 14, 16, 12, 8, 3, 2, 1, 0, 0, 0, 0, 6, 16, 20, 17, 8, 6, 2, 1, 1, 0, 0, 0, 0, 6, 20, 29, 25, 16, 10, 5, 2, 1, 1, 0, 0, 0, 0, 7, 24, 35, 36, 27, 14, 7, 6, 3, 1, 1, 0, 0, 0

Offset: 1

Author

Olivier Gérard, May 29 2024

Keywords

• nonn
• tabl

Comments

The corresponding irregular triangle (one entry for each partition of n) is A373241.

The sum of each row is A373243.

The corresponding triangle for sum of number of different parts is A092905.

The corresponding triangle for sum of number of different multiplicities is A373271.

Examples

			Array begins:
  0
  0,0
  0,1,0
  0,1,0,0
  0,2,0,0,0
  0,2,2,1,0,0
  0,3,2,1,0,0,0
  0,3,4,3,1,0,0,0
  0,4,6,4,2,2,0,0,0
  0,4,8,8,5,1,1,0,0,0
  ...
Example of computation:
T(9,3) = 6 because the partitions of 9 into 3 parts are
  7+1+1, 6+2+1, 5+3+1, 5+2+2, 4+4+1, 4+3+2, 3+3+3,
the numbers of different parts are
  2, 3, 3, 2, 2, 3, 1,
the numbers of different multiplicities are
  2, 1, 1, 2, 2, 1, 1,
the differences between them are
  0, 2, 2, 0, 0, 2, 0,
and the sum of these differences is 6.
		

Links

• Olivier Gérard, Table of n, a(n) for n = 1..820

Programs

Mathematica

Flatten[Table[
  Plus @@@
   Table[Map[Length[Union[#]] - Length[Union[Length /@ Split[#]]] &,
     IntegerPartitions[n, {k}]], {k, 1, n}], {n, 1, 20}]]