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A238325 Array: row n gives the number of occurrences of each possible antidiagonal partition of n, arranged in reverse-Mathematica order. 7
1, 2, 2, 1, 2, 3, 2, 2, 3, 2, 2, 6, 1, 2, 2, 4, 3, 4, 2, 2, 4, 6, 2, 6, 2, 2, 4, 4, 2, 3, 9, 4, 2, 2, 4, 4, 2, 6, 6, 3, 12, 1, 2, 2, 4, 4, 2, 4, 6, 3, 6, 6, 12, 5, 2, 2, 4, 4, 2, 4, 6, 6, 4, 6, 3, 18, 2, 4, 10, 2, 2, 4, 4, 2, 4, 6, 4, 4, 6, 3, 6, 12, 2, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Suppose that p is a partition of n, let F(p) be its Ferrers matrix, as defined at A237981, and let mXm be the size of F(p).  The numbers of 1's in each of the 2m-1 antidiagonals of F(p) form a partition of n.  Any partition which is associated with a partition of n in this manner is introduced here as an antidiagonal partition of n.  A000041(n) = sum of the numbers in row n; A000009(n) = number of terms in row n, since the antidiagonal partitions of n are the conjugates of the strict partitions of n.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..1000

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

EXAMPLE

The Mathematica ordering of the 6 antidiagonal partitions of 8 follows:  3221, 32111, 22211, 221111, 2111111, 11111111.)  Frequencies of these among the 22 partitions of 8 are given in reverse Mathematica ordering as follows:  11111111 occurs 2 times, 2111111 occurs 2 times, 221111 occurs 4 times, 22211 occurs 6 times, 32111 occurs 2 times, and 3221 occurs 6 times, so that row 8 of the array is 2 2 4 6 2 6.

...

First 12 rows:

1

2

2 1

2 3

2 2 3

2 2 6 1

2 2 4 3 4

2 2 4 6 2 6

2 2 4 4 2 3 9 1

2 2 4 4 2 6 6 3 12 1

2 2 4 4 2 4 6 3 6 6 12 5

2 2 4 4 2 4 6 6 4 6 3 18 2 4 10

MATHEMATICA

z = 20; ferrersMatrix[list_] := PadRight[Map[Table[1, {#}] &, #], {#, #} &[Max[#, Length[#]]]] &[list]; antiDiagPartNE[list_] := Module[{m = ferrersMatrix[list]}, Map[Diagonal[Reverse[m], #] &, Range[-#, #] &[Length[m] - 1]]]; a1[n_] :=  Last[Transpose[Tally[Map[DeleteCases[Reverse[Sort[Map[Count[#, 1] &, antiDiagPartNE[#]]]], 0] &, IntegerPartitions[n]]]]];

t = Table[a1[n], {n, 1, z}]; TableForm[Table[a1[n], {n, 1, z}]]   (* A238325, array *)

u = Flatten[t] (* A238325, sequence *)

(* Peter J. C. Moses, 18 February 2014 *)

CROSSREFS

Cf. A238326.

Sequence in context: A263765 A270073 A027348 * A238885 A023566 A090970

Adjacent sequences:  A238322 A238323 A238324 * A238326 A238327 A238328

KEYWORD

nonn,tabf,easy

AUTHOR

Clark Kimberling and Peter J. C. Moses, Feb 24 2014

STATUS

approved

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Last modified August 18 16:13 EDT 2017. Contains 290727 sequences.