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A230025 Triangular array:  t(n, k) = number of occurrences of k as the number of outliers in all the partitions of n. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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).

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.

Sequence in context: A217522 A287520 A130094 * A207869 A130210 A236459

Adjacent sequences:  A230022 A230023 A230024 * A230026 A230027 A230028

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Feb 23 2014

STATUS

approved

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Last modified August 16 13:28 EDT 2017. Contains 290623 sequences.