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

1,12

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.

Links

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

Example

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.

Mathematica

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

Crossrefs

Sequence in context: A125250 A048113 A028961 * A110177 A036273 A342563

Adjacent sequences: A373239 A373240 A373241 * A373243 A373244 A373245

Keyword

nonn,tabl

Author

Olivier Gérard, May 29 2024

Status

approved