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A238944 Number of partitions of n that have odd sized Ferrers matrix. 3

%I #7 Apr 06 2020 22:16:43

%S 0,2,1,3,2,6,7,13,14,23,26,40,47,69,85,119,145,198,242,320,391,507,

%T 620,794,968,1226,1493,1869,2269,2816,3408,4194,5056,6178,7423,9014,

%U 10793,13035,15561,18700,22251,26621

%N Number of partitions of n that have odd sized Ferrers matrix.

%C Also, the number of odd 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.

%F a(n) + A238945(n) = A000041(n).

%e (See the example at A238943.)

%t 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];

%t u = Table[Count[t[[n]], 0], {n, 1, z}] (* A238944 *)

%t v = Table[Count[t[[n]], 1], {n, 1, z}] (* A238945 *)

%Y Cf. A237981, A238945, A238943, A000041.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Mar 07 2014

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Last modified March 29 09:14 EDT 2024. Contains 371268 sequences. (Running on oeis4.)