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A238945 Number of partitions of n that have even-sized Ferrers matrix. 3

%I #8 Apr 07 2020 21:34:21

%S 1,0,2,2,5,5,8,9,16,19,30,37,54,66,91,112,152,187,248,307,401,495,635,

%T 781,990,1210,1517,1849,2296,2788,3434,4155,5087,6132,7460,8963,10844,

%U 12980,15624,18638,22332,26553

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

%C 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.

%F a(n) + A238944(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, A238944, A238943, A000041.

%K nonn,easy

%O 1,3

%A _Clark Kimberling_, Mar 07 2014

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Last modified March 29 09:32 EDT 2024. Contains 371268 sequences. (Running on oeis4.)