cp's OEIS Frontend

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

%I A238943 #19 Feb 25 2025 20:54:11
%S A238943 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,
%T A238943 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,
%U A238943 7,7,6,6,6,5,5,5,5,5,4,4,4,4,5,6,3,4
%N A238943 Triangular array read by rows: t(n,k) = size of the Ferrers matrix of p(n,k).
%C 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.
%F A238943 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.
%e A238943 First 8 rows:
%e A238943   1
%e A238943   2 2 2
%e A238943   3 2 3
%e A238943   4 3 2 3 4
%e A238943   5 4 3 3 3 4 5
%e A238943   6 5 4 4 3 3 4 3 4 5 6
%e A238943   7 6 5 5 4 4 4 3 3 4 5 4 5 6 7
%e A238943   8 7 6 6 5 5 5 4 4 4 4 5 3 4 4 5 6 4 5 6 7 8
%e A238943 For n = 3, the three partitions are [3], [2,1], [1,1,1]. Their respective Ferrers matrices derive from Ferrers graphs as follows:
%e A238943 The partition [3] has Ferrers graph 1 1 1, with Ferrers matrix of size 3:
%e A238943   1 1 1
%e A238943   0 0 0
%e A238943   0 0 0
%e A238943 The partition [2,1] has Ferrers graph
%e A238943   11
%e A238943   1
%e A238943 with Ferrers matrix of size 2:
%e A238943   1 1
%e A238943   1 0
%e A238943 The partition [1,1,1] has Ferrers graph
%e A238943   1
%e A238943   1
%e A238943   1
%e A238943 with Ferrers matrix of size 3
%e A238943   1 0 0
%e A238943   1 0 0
%e A238943   1 0 0
%e A238943 Thus row 3 is (3,2,3).
%t A238943 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]}]
%t A238943 u = TableForm[t]  (* A238943 array *)
%t A238943 v = Flatten[t]    (* A238943 sequence *)
%Y A238943 Cf. A238944, A238945, A237981, A000041.
%K A238943 nonn,tabf,easy
%O A238943 1,2
%A A238943 _Clark Kimberling_, Mar 07 2014