A238326 Array: row n gives the number of occurrences of each possible diagonal partition of n, arranged in reverse Mathematica order.

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

Offset: 1

Author

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

Keywords

• nonn
• tabf
• easy

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).

Examples

			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
		

Links

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

Crossrefs

Cf. A238325, A003114, A000041.

Programs

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 *)