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

1,2

Comments

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.

Links

Table of n, a(n) for n=1..86.

Formula

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.

Example

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).

Mathematica

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 *)

Crossrefs

Cf. A238944, A238945, A237981, A000041.

Sequence in context: A222334 A340716 A181948 * A070081 A366686 A071647

Adjacent sequences: A238940 A238941 A238942 * A238944 A238945 A238946

Keyword

nonn,tabf,easy

Author

Clark Kimberling, Mar 07 2014

Status

approved