A238004 Limiting row of the array at A238325.

2, 2, 4, 4, 2, 4, 6, 4, 4, 6, 4, 4, 12, 2, 4, 4, 8, 6, 8, 4, 4, 8, 12, 4, 12, 4, 4, 8, 8, 4, 6, 18, 8, 4, 4, 8, 8, 4, 12, 12, 6, 24, 2, 4, 4, 8, 8, 4, 8, 12, 6, 12, 12, 24, 10, 4, 4, 8, 8, 4, 8, 12, 12, 8, 12, 6, 36, 4, 8, 20, 4, 4, 8, 8, 4, 8, 12, 8, 8, 12

Offset

0,1

Comments

For fixed m >= 2 and sufficiently large k, the first m+1 antidiagonal partitions of k, listed in reverse Mathematica order, are as follows: p(0) = [1,1,...,1] (k 1's), p(1) = [2,1,...,1] (k-2 1's), p(2) = [2,2,1,...,1] (k-4 1's), ..., p[m] = [2,...,2,1,...,1] (m 2's and k-2m 1's). The number of occurrences of p(n) among all the partitions of k (for sufficiently large k), is a(n); see Example.

Links

Table of n, a(n) for n=0..79.

Clark Kimberling and Peter J. C. Moses, Ferrers Matrices and Related Partitions of Integers

Example

Referring to the antidiagonal partitions p(i) in Comments, p(0) occurs 2 times for all k >=2; p(1) occurs 2 times for all k >=5; p(2) occurs 4 times for all k >= 7; p(3) occurs 4 times for all k >= 9; etc., so that A238004 begins with 2, 2, 4, 4.

Mathematica

ferrersMatrix[list_] := PadRight[Map[Table[1, {#}] &, #], {#, #} &[Max[#, Length[#]]]] &[list]; antiDiagPartNE[list_] := Module[{m = ferrersMatrix[list]},    Map[Diagonal[Reverse[m], #] &, Range[-#, #] &[Length[m] - 1]]]; a[n_] := Last[Transpose[Tally[Map[DeleteCases[Reverse[Sort[Map[Count[#, 1] &, antiDiagPartNE[#]]]], 0] &, IntegerPartitions[n]]]]]
Take[a[40], 100] (* Peter J. C. Moses, Feb 18 2014 *)

Crossrefs

Cf. A238325.

Sequence in context: A060267 A214516 A352502 * A048244 A056673 A353761

Adjacent sequences: A238001 A238002 A238003 * A238005 A238006 A238007

Keyword

nonn,easy

Author

Clark Kimberling and Peter J. C. Moses, Feb 25 2014

Status

approved