A373269 -id:A373269 - cp's OEIS frontend

A373241 T(n,k) is the difference between the number of different parts and the number of different multiplicities in the k-th partition of n in graded reverse lexicographic ordering (A080577).

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 2, 0, 0, 2, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 2, 0, 1, 2, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 2, 0, 1, 2, 0, 1, 0, 0, 2, 2, 1, 1, 1, 0, 0, 1, 0, 3, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Olivier Gérard, May 29 2024

This difference is always nonnegative.
The number of zero values in each row is A098859.
The number of ones in each row is A325244.
The number of positive entries in each row is A336866.
The corresponding regular triangle for partitions of n of length k is A373242.
The sum of each row is A373243.

			The array begins
  0
  0,0
  0,1,0
  0,1,0,0,0
  0,1,1,0,0,0,0
  0,1,1,0,0,2,0,0,1,0,0
  0,1,1,0,1,2,0,0,0,1,0,0,0,0,0
  0,1,1,0,1,2,0,0,2,0,1,0,0,1,1,1,0,0,0,0,0,0
  0,1,1,0,1,2,0,1,2,0,1,0,0,2,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,0
  ...

Olivier Gérard, Table of n, a(n) for n = 1..215307

Cf. A373269 a triangle of the same shape and order for number of multiplicities.

Cf. A080577, A098859, A325244, A336866, A373242, A373243.

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

A373271 a(n) = sum for all integer partitions of n of the number of distinct multiplicities in each partition.

Original entry on oeis.org

1, 2, 3, 6, 10, 14, 24, 34, 49, 70, 103, 134, 195, 258, 347, 461, 624, 796, 1066, 1358, 1763, 2250, 2903, 3631, 4644, 5805, 7309, 9083, 11381, 13998, 17428, 21369, 26336, 32174, 39451, 47847, 58399, 70610, 85590, 103077, 124462, 149169, 179368, 214300, 256397

Offset: 1

Olivier Gérard, May 29 2024

nonn

Sum of the rows of A373269.
Sum of the rows of A373270.
The multiplicity of a part in an integer partition (or composition) is the number of times it appears in the partition, seen as a list.
The multiplicity of 3 in the partition 12 = 5+3+3+1 is 2.
For this sequence, only distinct multiplicities appearing for parts of the partition are counted, only once for a given partition.
If all multiplicities of all parts of all integer partitions of n are counted, one gets A000070 (1, 2, 4, 7, 12, 19, 30, 45, 67, 97, ...).
If all distinct multiplicities of all parts of all integer partitions are summed, one gets A373273 (1, 3, 5, 11, 18, 29, 48, 74, 107, 161, ...).
If all multiplicities of all parts of all integer partitions of n are summed, one gets A006128 (1, 3, 6, 12, 20, 35, 54, 86, 128, 192, ...).

			Example for n=20:
the partition 20=4+3+3+3+3+2+1+1
has multiplicities 1, 4, 1, 2
for the parts 4,3,2,1 listed in descending order.
It has 3 different multiplicities (1, 2 and 4) and contributes 3 to a(20) = 1358.

Alois P. Heinz, Table of n, a(n) for n = 1..200

Cf. A373243, A098859.

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

A373270 Triangle read by rows: T(n,k) is the sum for all integer partitions of n of length k of the number of different multiplicities.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 4, 3, 2, 1, 1, 3, 7, 6, 4, 2, 1, 1, 4, 8, 8, 6, 4, 2, 1, 1, 4, 10, 12, 10, 5, 4, 2, 1, 1, 5, 12, 15, 13, 11, 6, 4, 2, 1, 1, 5, 15, 21, 20, 17, 11, 6, 4, 2, 1, 1, 6, 16, 25, 26, 21, 16, 10, 6, 4, 2, 1, 1, 6, 20, 33, 36, 34, 24, 17, 11, 6, 4, 2, 1, 1, 7, 22, 38, 46, 44, 34, 25, 17, 11, 6, 4, 2, 1, 1, 7, 25, 48, 58, 56, 50, 38, 24, 16, 11, 6, 4, 2, 1

Offset: 1

Olivier Gérard, May 29 2024

			Array begins:
  1,
  1, 1,
  1, 1,  1,
  1, 2,  2,  1,
  1, 2,  4,  2,  1,
  1, 3,  4,  3,  2,  1,
  1, 3,  7,  6,  4,  2,  1,
  1, 4,  8,  8,  6,  4,  2, 1,
  1, 4, 10, 12, 10,  5,  4, 2, 1,
  1, 5, 12, 15, 13, 11,  6, 4, 2, 1,
  1, 5, 15, 21, 20, 17, 11, 6, 4, 2, 1,
  ...
Example of computation:
T(9,3) = 10 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 number of different multiplicities are
  2, 1, 1, 2, 2, 1, 1,
and the sum of these multiplicities is 10.

Alois P. Heinz, Rows n = 0..200, flattened (first 40 rows from Olivier Gérard)

Cf. A373242, A373269, A373271.

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

cp's OEIS Frontend

A373241 T(n,k) is the difference between the number of different parts and the number of different multiplicities in the k-th partition of n in graded reverse lexicographic ordering (A080577).

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A373271 a(n) = sum for all integer partitions of n of the number of distinct multiplicities in each partition.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A373270 Triangle read by rows: T(n,k) is the sum for all integer partitions of n of length k of the number of different multiplicities.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica