A230025 Triangular array: t(n, k) = number of occurrences of k as the number of outliers in all the partitions of n.

1, 0, 2, 1, 0, 2, 1, 2, 0, 2, 1, 2, 2, 0, 2, 1, 2, 4, 2, 0, 2, 1, 4, 2, 4, 2, 0, 2, 2, 2, 6, 2, 6, 2, 0, 2, 2, 6, 2, 8, 2, 6, 2, 0, 2, 2, 4, 12, 2, 8, 2, 8, 2, 0, 2, 2, 8, 6, 14, 2, 10, 2, 8, 2, 0, 2, 3, 6, 14, 8, 18, 2, 10, 2, 10, 2, 0, 2, 3, 10, 10, 20, 10

Offset: 1

Author

Clark Kimberling, Feb 23 2014

Keywords

• nonn
• tabl
• easy

Comments

Definitions: the self-conjugate portion of a partition p is the portion of the Ferrers graph of p that remains unchanged when p is reflected about its principal diagonal. The outliers of p are the nodes of the Ferrers graph that lie outside the self-conjugate portion of p.

Sum of numbers in row n is A000041(n).

t(n,k) is the number of partitions p of n such that d(p,p*) = k, where d is the distance function introduced in A366156 and p* is the conjugate of p. - Clark Kimberling, Oct 03 2023

Examples

			The first 9 rows:
  1
  0 2
  1 0 2
  1 2 0 2
  1 2 2 0 2
  1 2 4 2 0 2
  1 4 2 4 2 0 2
  2 2 6 2 6 2 0 2
  2 6 2 8 2 6 2 0 2
The Ferrers graph of the partition p = [4,4,1,1] of 10 follows:
  1 1 1 1
  1 1 1 1
  1
  1
The self-conjugate portion of p is
  1 1 1 1
  1 1
  1
  1
so that the number of outliers of p is 2.
		

Crossrefs

Cf. A000041, A366156.

Programs

Mathematica

ferrersMatrix[list_] := PadRight[Map[Table[1, {#}] &, #], {#, #} &[Max[#, Length[#]]]] &[list]; conjugatePartition[part_] := Table[Count[#, ?(# >= i &)], {i, First[#]}] &[part]; selfConjugatePortion[list] := ferrersMatrix[#]*ferrersMatrix[conjugatePartition[#]] &[list]; outliers[list_] := Count[Flatten[ferrersMatrix[#] - selfConjugatePortion[#] &[list]], 1]; a[n_] := Map[outliers, IntegerPartitions[n]]; t = Table[Count[a[n], k], {n, 1, 13}, {k, 0, n - 1}]
u = Flatten[t]
(* Peter J. C. Moses, Feb 21 2014 *)