A239514 - OEIS

%I #6 Jan 28 2022 01:02:52

%S 0,0,0,0,1,1,0,2,2,3,2,4,2,7,6,7,6,10,7,14,12,18,12,22,18,23,23,31,29,

%T 42,33,45,42,54,49,68,62,78,76,95,87,110,102,124,128,150,141,178,174,

%U 203,203,237,228,272,269,308,318,360,356,422,420,472,482

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

%C 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.

%F a(12) counts these partitions: 615, 642, 43131, 3121212.

%t z = 75; Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Min[p]] == Length[p] - 1], {n, 1, z}]  (* A239510 *)

%t Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p] - 1], {n, 1, z}]  (* A239511 *)

%t Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p]] == Length[p] - 1], {n, 1, z}]  (* A237828 *)

%t Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Length[p]] == Length[p] - 1], {n, 1, z}]  (* A239513 *)

%t Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p] - 1], {n, 1, z}] (* A239514 *)

%Y Cf. A239510, A239511, A237828, A239513, A239482.

%K nonn,easy

%O 1,8

%A _Clark Kimberling_, Mar 24 2014