cp's OEIS Frontend

A239500 Number of partitions p of n such that if h = (number of parts of p), then h is an (h,1)-separator of p; see Comments.

%I A239500 #13 Jan 28 2022 01:00:18
%S A239500 0,0,1,0,1,1,1,1,1,2,2,3,2,3,3,4,4,5,5,6,7,8,9,11,12,13,15,16,18,20,
%T A239500 22,24,27,29,32,36,39,43,48,53,58,65,70,78,85,93,101,112,120,132,143,
%U A239500 156,168,184,198,216,233,253,273,298,320,348,376,407,439
%N A239500 Number of partitions p of n such that if h = (number of parts of p), then h is an (h,1)-separator of p; see Comments.
%C A239500 Suppose that p is a partition of n into 2 or more parts and that h is a part of p. Then p is (h,0)-separable if there is an ordering x, h, x, h, ..., h, x of the parts of p, where each x represents any part of p except h. Here, the number of h's on the ends of the ordering is 0. Similarly, p is (h,1)-separable if there is an ordering x, h, x, h, ..., x, h, where the number of h's on the ends is 1; next, p is (h,2)-separable if there is an ordering h, x, h, ..., x, h. Finally, p is h-separable if it is (h,i)-separable for i = 0,1,2.
%e A239500 a(12) counts these partitions: 84, 4431, 4422.
%t A239500 z = 35; t1 = Table[Count[IntegerPartitions[n],  p_ /; 2 Count[p, Min[p]] == Length[p]], {n, 1, z}]  (* A239497 *)
%t A239500 t2 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p]], {n, 1, z}] (* A239498 *)
%t A239500 t3 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p]] == Length[p]], {n, 1, z}] (* A118096 *)
%t A239500 t4 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Length[p]] == Length[p]], {n, 1, z}] (* A239500 *)
%t A239500 t5 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p]], {n, 1, z}]  (* A239501 *)
%Y A239500 Cf. A239497, A239498, A118096, A239501, A239482.
%K A239500 nonn,easy
%O A239500 1,10
%A A239500 _Clark Kimberling_, Mar 24 2014