A238944 -id:A238944 - cp's OEIS frontend

A238943 Triangular array read by rows: t(n,k) = size of the Ferrers matrix of p(n,k).

1, 2, 2, 3, 2, 3, 4, 3, 2, 3, 4, 5, 4, 3, 3, 3, 4, 5, 6, 5, 4, 4, 3, 3, 4, 3, 4, 5, 6, 7, 6, 5, 5, 4, 4, 4, 3, 3, 4, 5, 4, 5, 6, 7, 8, 7, 6, 6, 5, 5, 5, 4, 4, 4, 4, 5, 3, 4, 4, 5, 6, 4, 5, 6, 7, 8, 9, 8, 7, 7, 6, 6, 6, 5, 5, 5, 5, 5, 4, 4, 4, 4, 5, 6, 3, 4

Offset: 1

Clark Kimberling, Mar 07 2014

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.

			First 8 rows:
  1
  2 2 2
  3 2 3
  4 3 2 3 4
  5 4 3 3 3 4 5
  6 5 4 4 3 3 4 3 4 5 6
  7 6 5 5 4 4 4 3 3 4 5 4 5 6 7
  8 7 6 6 5 5 5 4 4 4 4 5 3 4 4 5 6 4 5 6 7 8
For n = 3, the three partitions are [3], [2,1], [1,1,1]. Their respective Ferrers matrices derive from Ferrers graphs as follows:
The partition [3] has Ferrers graph 1 1 1, with Ferrers matrix of size 3:
  1 1 1
  0 0 0
  0 0 0
The partition [2,1] has Ferrers graph
  11
  1
with Ferrers matrix of size 2:
  1 1
  1 0
The partition [1,1,1] has Ferrers graph
  1
  1
  1
with Ferrers matrix of size 3
  1 0 0
  1 0 0
  1 0 0
Thus row 3 is (3,2,3).

Cf. A238944, A238945, A237981, A000041.

p[n_, k_] := p[n, k] = IntegerPartitions[n][[k]]; a[t_] := Max[Max[t], Length[t]]; t = Table[a[p[n, k]], {n, 1, 10}, {k, 1, PartitionsP[n]}]
u = TableForm[t]  (* A238943 array *)
v = Flatten[t]    (* A238943 sequence *)

t(n,k) = max{max(p(n,k)), length(p(n,k))}, where p(n,k) is the k-th partition of n in Mathematica order.

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

Original entry on oeis.org

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

Clark Kimberling, Mar 07 2014

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.

			(See the example at A238943.)

Cf. A237981, A238944, A238943, A000041.

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 *)

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

cp's OEIS Frontend

A238943 Triangular array read by rows: t(n,k) = size of the Ferrers matrix of p(n,k).

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