cp's OEIS Frontend

A239456 Number of partitions p of n such that if h = max(p), then h is an (h,2)-separator of p; see Comments.

%I A239456 #6 Jan 28 2022 01:11:27
%S A239456 0,0,0,0,1,0,1,2,1,1,4,2,3,4,4,5,7,5,7,9,9,9,14,11,13,16,17,19,22,20,
%T A239456 25,28,29,30,38,37,41,45,48,51,60,59,67,73,76,82,93,94,103,111,121,
%U A239456 127,142,143,158,171,180,191,211,218,236,252,270,284,309,320
%N A239456 Number of partitions p of n such that if h = max(p), then h is an (h,2)-separator of p; see Comments.
%C A239456 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 A239456 a(11) counts these 4 partitions: 515, 434, 31313, 2121212.
%t A239456 z = 75; Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Min[p]] == Length[p] + 1], {n, 1, z}]  (* A239729 *)
%t A239456 Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p] + 1], {n, 1, z}] (* A239481 *)
%t A239456 Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Max[p]] == Length[p] + 1], {n, 1, z}] (* A239456 *)
%t A239456 Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Length[p]] == Length[p] + 1], {n, 1, z}] (* A239499 *)
%t A239456 Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p] + 1], {n, 1, z}]  (*A239689 *)
%Y A239456 Cf. A239729, A239481, A239499, A239689.
%K A239456 nonn,easy
%O A239456 1,8
%A A239456 _Clark Kimberling_, Mar 25 2014