A373241 -id:A373241 - cp's OEIS frontend

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

Olivier Gérard, May 29 2024

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.

			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.

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

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

A373243 a(n) = sum for all integer partitions of n of the difference between number of different parts and number of different multiplicities.

Original entry on oeis.org

0, 0, 1, 1, 2, 5, 6, 11, 18, 27, 36, 61, 77, 115, 161, 223, 291, 416, 531, 729, 951, 1256, 1605, 2132, 2694, 3491, 4423, 5659, 7079, 9027, 11201, 14102, 17484, 21789, 26822, 33309, 40734, 50160, 61195, 74893, 90846, 110722, 133697, 162026, 195104, 235244

Offset: 1

Olivier Gérard, May 29 2024

nonn

Sum of the rows of A373241 or A373242.

			From the eighth row of A373241: a(8)=11
  0, 1, 1, 0, 1, 2, 0, 0, 2, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0
or the tenth row of A373242: a(10)=27
  0, 4, 8, 8, 5, 1, 1, 0, 0, 0

Cf. A373241, A373242.

Table[Plus @@
  Table[Plus @@
    Map[Length[Union[#]] - Length[Union[Length /@ Split[#]]] &,
     IntegerPartitions[n, {k}]], {k, 1, n}], {n, 1, 40}]

A373244 T(n,k) = number of integer partitions of n into k parts for which the number of distinct parts is equal to the number of distinct multiplicities.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, 3, 2, 2, 1, 1, 1, 1, 3, 2, 2, 2, 1, 1, 1, 0, 4, 2, 3, 1, 2, 1, 1, 1, 1, 4, 3, 2, 4, 2, 2, 1, 1, 1, 0, 5, 3, 4, 5, 4, 2, 2, 1, 1, 1, 1, 5, 3, 3, 5, 4, 3, 2, 2, 1, 1, 1, 0, 6, 4, 5, 8, 6, 5, 4, 2, 2, 1, 1, 1, 1, 6, 4, 5, 10, 6, 7, 5, 4, 2, 2, 1, 1

Offset: 1

Olivier Gérard, May 29 2024

Row sum is A098859 (Wilf partitions of n).

Counts the zeros in A373241 or A373242.

			Array begins:
  1,
  1, 1,
  1, 0, 1,
  1, 1, 1, 1,
  1, 0, 2, 1, 1,
  1, 1, 2, 1, 1, 1,
  1, 0, 3, 2, 2, 1, 1,
  1, 1, 3, 2, 2, 2, 1, 1,
  1, 0, 4, 2, 3, 1, 2, 1, 1
  ...

See references listed in A098859.

Alois P. Heinz, Rows n = 1..200, flattened (first 40 rows from Olivier Gérard)

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

cp's OEIS Frontend

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.

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A373243 a(n) = sum for all integer partitions of n of the difference between number of different parts and number of different multiplicities.

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A373244 T(n,k) = number of integer partitions of n into k parts for which the number of distinct parts is equal to the number of distinct multiplicities.

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Author

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Examples

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Mathematica