cp's OEIS Frontend

A239518 Number of partitions p of n that are separable by the number of parts of p; see Comments.

%I A239518 #8 Jan 28 2022 00:57:59
%S A239518 0,0,1,0,2,2,3,3,3,5,6,7,8,9,11,13,15,17,20,22,25,28,32,36,42,45,52,
%T A239518 57,65,71,81,88,100,109,122,134,149,162,180,197,218,238,262,286,315,
%U A239518 343,376,410,449,488,534,580,633,687,749,812,883,956,1038,1123
%N A239518 Number of partitions p of n that are separable by the number of parts of p; see Comments.
%C A239518 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 A239518 Let h = number of parts of p. The (h,0)-separable partition of 11 are 92, 731, 632, 434; the (h,1)-separable partition is 2414; the (h,2)-partition is 353. So, there are 4 + 1 + 1 = 6 h-separable partitions of 11.
%t A239518 z = 75; t1 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, Min[p]] <= Length[p] + 1], {n, 1, z}]  (* A239515 *)
%t A239518 t2 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 2*Min[p]] <= Length[p] + 1], {n, 1, z}]  (* A239516 *)
%t A239518 t3 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, Max[p]] <= Length[p] + 1], {n, 1, z}]  (* A239517 *)
%t A239518 t4 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, Length[p]] <= Length[p] + 1], {n, 1, z}]  (* A239518 *)
%Y A239518 Cf. A239515, A239516, A239517, A239482.
%K A239518 nonn,easy
%O A239518 1,5
%A A239518 _Clark Kimberling_, Mar 26 2014