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A238943 Triangular array read by rows: t(n,k) = size of the Ferrers matrix of p(n,k). 3
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 (list; graph; refs; listen; history; text; internal format)
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 1s as nodes, and pad the graph with 0s 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: A221999 A222334 A181948 * A070081 A034883 A071647

Adjacent sequences:  A238940 A238941 A238942 * A238944 A238945 A238946

KEYWORD

nonn,tabf,easy

AUTHOR

Clark Kimberling, Mar 07 2014

STATUS

approved

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Last modified June 28 09:49 EDT 2017. Contains 288813 sequences.