cp's OEIS Frontend

A239510 Number of partitions p of n such that if h = min(p), then h is an (h,0)-separator of p; see Comments.

%I A239510 #8 Jan 28 2022 01:01:17
%S A239510 0,0,0,0,1,1,2,4,5,7,11,13,18,24,30,37,48,59,73,90,109,132,163,193,
%T A239510 233,280,334,397,475,559,663,784,924,1085,1279,1494,1751,2049,2392,
%U A239510 2784,3248,3769,4382,5081,5887,6808,7879,9087,10486,12083,13910,15988,18384
%N A239510 Number of partitions p of n such that if h = min(p), then h is an (h,0)-separator of p; see Comments.
%C A239510 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 A239510 a(9) counts these 5 partitions: 612, 513, 414, 423, 312121.
%t A239510 z = 75; Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Min[p]] == Length[p] - 1], {n, 1, z}]  (* A239510 *)
%t A239510 Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p] - 1], {n, 1, z}]  (* A239511 *)
%t A239510 Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p]] == Length[p] - 1], {n, 1, z}]  (* A237828 *)
%t A239510 Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Length[p]] == Length[p] - 1], {n, 1, z}]  (* A239513 *)
%t A239510 Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p] - 1], {n, 1, z}] (* A239514 *)
%Y A239510 Cf. A239511, A237828, A239513, A239514, A239482.
%K A239510 nonn,easy
%O A239510 1,7
%A A239510 _Clark Kimberling_, Mar 24 2014