A238325 Array: row n gives the number of occurrences of each possible antidiagonal partition of n, arranged in reverse-Mathematica order.

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

Offset: 1

Author

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

Examples

			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,  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;
		

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

Crossrefs

Cf. A238326.

Programs

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 *)

Extensions

Example corrected by Peter J. Taylor, Apr 10 2022