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

1,3

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

Links

Table of n, a(n) for n=1..83.

Example

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.

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 *)

Crossrefs

Cf. A000041, A366156.

Sequence in context: A130094 A337005 A366745 * A330374 A207869 A130210

Adjacent sequences: A230022 A230023 A230024 * A230026 A230027 A230028

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Feb 23 2014

Status

approved