A238945 Number of partitions of n that have even-sized Ferrers matrix.

1, 0, 2, 2, 5, 5, 8, 9, 16, 19, 30, 37, 54, 66, 91, 112, 152, 187, 248, 307, 401, 495, 635, 781, 990, 1210, 1517, 1849, 2296, 2788, 3434, 4155, 5087, 6132, 7460, 8963, 10844, 12980, 15624, 18638, 22332, 26553

Offset: 1

Author

Clark Kimberling, Mar 07 2014

Keywords

• nonn
• easy

Comments

Also, the number of even numbers in row n of the array at A238943. Suppose that p is a partition of n, and let m = max{greatest part of p, number of parts of p}. Write the Ferrers graph of p with 1's as nodes, and pad the graph with 0's to form an m X m square matrix, which is introduced at A237981 as the Ferrers matrix of p, denoted by f(p). The size of f(p) is m.

Examples

			(See the example at A238943.)
		

Crossrefs

Cf. A237981, A238944, A238943, A000041.

Programs

Mathematica

p[n_, k_] := p[n, k] = IntegerPartitions[n][[k]]; a[t_] := Max[Max[t], Length[t]]; z = 42; t = Mod[Table[a[p[n, k]], {n, 1, z}, {k, 1, PartitionsP[n]}], 2];
u = Table[Count[t[[n]], 0], {n, 1, z}]  (* A238944 *)
v = Table[Count[t[[n]], 1], {n, 1, z}]  (* A238945 *)

Formula

a(n) + A238944(n) = A000041(n).