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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.
5
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
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
KEYWORD
nonn,tabl
AUTHOR
Olivier Gérard, May 29 2024
STATUS
approved