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A238326 Array: row n gives the number of occurrences of each possible diagonal partition of n, arranged in reverse Mathematica order. 5
1, 2, 3, 4, 1, 5, 2, 6, 3, 2, 7, 4, 4, 8, 5, 6, 3, 9, 6, 8, 6, 1, 10, 7, 10, 9, 4, 2, 11, 8, 12, 12, 8, 3, 2, 12, 9, 14, 15, 12, 5, 4, 4, 2, 13, 10, 16, 18, 16, 10, 5, 6, 3, 4, 14, 11, 18, 21, 20, 15, 6, 6, 8, 6, 6, 4, 15, 12, 20, 24, 24, 20, 7, 12, 10, 9, 8 (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 1s in each of the 2m-1 diagonals 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 a diagonal partition of n.  A000041(n) = sum of the numbers in row n; A003114(n) = number of terms in row n.  Every diagonal partition is an antidiagonal partition, as in A238325 (but not conversely).

LINKS

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

EXAMPLE

The Mathematica ordering of the 3 antidiagonal partitions of 6 follows:  2211, 21111, 111111.)  Frequencies of these among the 11 partitions of 6 are given in reverse Mathematica ordering as follows:  111111 occurs 6 times, 21111 occurs 3 times, and 2211 occurs 2 times, so that row 6 of the array is 6 3 2.

...

First 9 rows:

1

2

3

4 1

5 2

6 3 2

7 4 4

8 5 6 3

9 6 8 6 1

MATHEMATICA

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

Map[Last[Transpose[Tally[Map[DeleteCases[Reverse[Sort[Map[Count[#, 1] &, diagPartSE[#]]]], 0] &, IntegerPartitions[#]]]]] &, Range[z]] // TableForm

(* Peter J. C. Moses, Feb 25 2014 *)

CROSSREFS

Cf. A238325, A003114, A000041.

Sequence in context: A097150 A278108 A087165 * A083480 A179547 A023133

Adjacent sequences:  A238323 A238324 A238325 * A238327 A238328 A238329

KEYWORD

nonn,tabf,easy

AUTHOR

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

STATUS

approved

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Last modified May 23 19:46 EDT 2017. Contains 286926 sequences.