cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 4, 5, 7, 11, 13, 18, 24, 30, 37, 48, 59, 73, 90, 109, 132, 163, 193, 233, 280, 334, 397, 475, 559, 663, 784, 924, 1085, 1279, 1494, 1751, 2049, 2392, 2784, 3248, 3769, 4382, 5081, 5887, 6808, 7879, 9087, 10486, 12083, 13910, 15988, 18384
Offset: 1

Views

Author

Clark Kimberling, Mar 24 2014

Keywords

Comments

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.

Examples

			a(9) counts these 5 partitions: 612, 513, 414, 423, 312121.

Crossrefs

Cf. A239511, A237828, A239513, A239514, A239482.

Programs

  • Mathematica
    z = 75; Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Min[p]] == Length[p] - 1], {n, 1, z}]  (* A239510 *)
Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p] - 1], {n, 1, z}]  (* A239511 *)
Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p]] == Length[p] - 1], {n, 1, z}]  (* A237828 *)
Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Length[p]] == Length[p] - 1], {n, 1, z}]  (* A239513 *)
Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p] - 1], {n, 1, z}] (* A239514 *)

A239513 Number of partitions p of n such that if h = (number of parts of p), then h is an (h,0)-separator of p; see Comments.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 9, 10, 12, 13, 15, 17, 19, 21, 25, 27, 31, 35, 40, 44, 50, 55, 62, 68, 76, 83, 93, 101, 112, 122, 136, 147, 163, 177, 196, 213, 235, 255, 281, 305, 335, 363, 398, 431, 471, 510, 556, 601, 654, 706, 768, 828
Offset: 1

Views

Author

Clark Kimberling, Mar 24 2014

Keywords

Comments

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.

Examples

			a(13) counts these 5 partitions: 931, 832, 634, 535, 15151.

Crossrefs

Cf. A239510, A239511, A238928, A239514, A239482.

Programs

  • Mathematica
    z = 75; Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Min[p]] == Length[p] - 1], {n, 1, z}]  (* A239510 *)
Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p] - 1], {n, 1, z}]  (* A239511 *)
Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p]] == Length[p] - 1], {n, 1, z}]  (* A237828 *)
Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Length[p]] == Length[p] - 1], {n, 1, z}]  (* A239513 *)
Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p] - 1], {n, 1, z}] (* A239514 *)

A239514 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.

Original entry on oeis.org

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, 42, 33, 45, 42, 54, 49, 68, 62, 78, 76, 95, 87, 110, 102, 124, 128, 150, 141, 178, 174, 203, 203, 237, 228, 272, 269, 308, 318, 360, 356, 422, 420, 472, 482
Offset: 1

Views

Author

Clark Kimberling, Mar 24 2014

Keywords

Comments

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.

Crossrefs

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

Programs

  • Mathematica
    z = 75; Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Min[p]] == Length[p] - 1], {n, 1, z}]  (* A239510 *)
Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p] - 1], {n, 1, z}]  (* A239511 *)
Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p]] == Length[p] - 1], {n, 1, z}]  (* A237828 *)
Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Length[p]] == Length[p] - 1], {n, 1, z}]  (* A239513 *)
Table[Count[Rest[IntegerPartitions[n]], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p] - 1], {n, 1, z}] (* A239514 *)

Formula

a(12) counts these partitions: 615, 642, 43131, 3121212.
Showing 1-3 of 3 results.

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