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

Author

Olivier Gérard, May 29 2024

Keywords

• nonn
• tabf

Comments

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.

Examples

			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
  ...
		

Links

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

Crossrefs

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

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

Programs

Mathematica

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