cp's OEIS Frontend

A239513 Number of partitions p of n such that if h = (number of parts of p), then h is an (h,0)-separator of p; see Comments.

%I A239513 #6 Jan 28 2022 01:02:26
%S A239513 0,0,0,0,1,1,1,1,2,2,3,3,5,5,7,8,9,10,12,13,15,17,19,21,25,27,31,35,
%T A239513 40,44,50,55,62,68,76,83,93,101,112,122,136,147,163,177,196,213,235,
%U A239513 255,281,305,335,363,398,431,471,510,556,601,654,706,768,828
%N A239513 Number of partitions p of n such that if h = (number of parts of p), then h is an (h,0)-separator of p; see Comments.
%C A239513 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 A239513 a(13) counts these 5 partitions: 931, 832, 634, 535, 15151.
%t A239513 z = 75; Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Min[p]] == Length[p] - 1], {n, 1, z}]  (* A239510 *)
%t A239513 Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p] - 1], {n, 1, z}]  (* A239511 *)
%t A239513 Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p]] == Length[p] - 1], {n, 1, z}]  (* A237828 *)
%t A239513 Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Length[p]] == Length[p] - 1], {n, 1, z}]  (* A239513 *)
%t A239513 Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p] - 1], {n, 1, z}] (* A239514 *)
%Y A239513 Cf. A239510, A239511, A238928, A239514, A239482.
%K A239513 nonn,easy
%O A239513 1,9
%A A239513 _Clark Kimberling_, Mar 24 2014