cp's OEIS Frontend

A239689 Number of partitions p of n such that if h = max(p) - min(p), then h is an (h,2)-separator of p; see Comments.

%I A239689 #7 Jan 28 2022 01:13:30
%S A239689 0,0,0,1,0,0,1,1,0,2,0,1,3,2,1,4,0,4,5,5,4,5,5,7,7,10,6,15,10,13,13,
%T A239689 14,16,20,18,26,24,29,28,34,27,39,40,47,46,57,55,65,63,75,72,83,85,
%U A239689 101,101,117,115,136,134,157,156,170,181,200,197,229,230
%N A239689 Number of partitions p of n such that if h = max(p) - min(p), then h is an (h,2)-separator of p; see Comments.
%C A239689 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, or 2.
%e A239689 a(13) counts these 3 partitions: 24232, 2321212, 121212121.
%t A239689 z = 75; Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Min[p]] == Length[p] + 1], {n, 1, z}] (* A239729 *)
%t A239689 Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p] + 1], {n, 1, z}] (* A239481 *)
%t A239689 Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Max[p]] == Length[p] + 1], {n, 1, z}] (* A239456 *)
%t A239689 Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Length[p]] == Length[p] + 1], {n, 1, z}] (* A239499 *)
%t A239689 Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p] + 1], {n, 1, z}] (* A239689 *)
%Y A239689 Cf. A239729, A239481, A239456, A239499.
%K A239689 nonn,easy
%O A239689 1,10
%A A239689 _Clark Kimberling_, Mar 25 2014