A239467 -id:A239467 - cp's OEIS frontend

A239482 Number of (2,0)-separable partitions of n; see Comments.

0, 1, 0, 1, 2, 2, 3, 5, 5, 7, 10, 11, 14, 19, 21, 27, 34, 39, 48, 60, 69, 84, 102, 119, 142, 172, 199, 237, 282, 328, 387, 458, 530, 623, 730, 847, 987, 1153, 1331, 1547, 1796, 2071, 2394, 2771, 3183, 3671, 4227, 4849, 5568, 6395, 7313, 8377, 9584, 10940

Offset: 3

Clark Kimberling, Mar 20 2014

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.

			The (2,0)-separable partitions of 10 are 721, 523, 424, 42121, 1212121, so that a(10) = 5.

Cf. A239467, A165652, A239483, A239484, A239485.

z = 65; -1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 1] == Length[p] - 1], {n, 2, z}]  (* A165652 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2] == Length[p] - 1], {n, 3, z}]  (* A239482 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 3] == Length[p] - 1], {n, 4, z}]  (* A239483 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 4] == Length[p] - 1], {n, 5, z}]  (* A239484 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 5] == Length[p] - 1], {n, 6, z}] (* A239485 *)

A239468 Number of 2-separable partitions of n; see Comments.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 4, 6, 7, 10, 12, 16, 20, 25, 31, 39, 47, 59, 71, 87, 105, 128, 153, 185, 221, 265, 315, 377, 445, 530, 625, 739, 870, 1025, 1201, 1411, 1649, 1930, 2249, 2625, 3050, 3549, 4116, 4773, 5523, 6391, 7375, 8515, 9806, 11293, 12980, 14917, 17110

Offset: 1

Clark Kimberling, Mar 20 2014

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.

			(2,0)-separable partitions of 7: 421, 12121;
(2,1)-separable partitions of 7: 52;
(2,2)-separable partitions of 7: 232;
2-separable partitions of 7: 421, 12121, 52, 232, so that a(7) = 4.

Cf. A239467, A239469, A239470, A239471.

z = 55; t1 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 1] <= Length[p] + 1], {n, 1, z}] (* A239467 *)
t2 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 2] <= Length[p] + 1], {n, 1, z}] (* A239468 *)
t3 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 3] <= Length[p] + 1], {n, 1, z}] (* A239469 *)
t4 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 4] <= Length[p] + 1], {n, 1, z}] (* A239470 *)
t5 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 5] <= Length[p] + 1], {n, 1, z}] (* A239472 *)

A239469 Number of 3-separable partitions of n; see Comments.

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 3, 4, 5, 6, 8, 11, 13, 15, 20, 24, 30, 35, 43, 52, 63, 74, 89, 106, 127, 148, 177, 208, 246, 287, 338, 396, 464, 538, 630, 732, 853, 985, 1145, 1324, 1532, 1765, 2038, 2345, 2702, 3098, 3562, 4081, 4679, 5348, 6120, 6987, 7978, 9087, 10359

Offset: 1

Clark Kimberling, Mar 20 2014

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.

			(3,0)-separable partitions of 7: 232;
(3,1)-separable partitions of 7: 43;
(3,2)-separable partitions of 7: 3231;
3-separable partitions of 7: 232, 43, 3231, so that a(7) = 3.

Cf. A239467, A239468, A239470, A239471.

z = 55; t1 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 1] <= Length[p] + 1], {n, 1, z}] (* A239467 *)
t2 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 2] <= Length[p] + 1], {n, 1, z}] (* A239468 *)
t3 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 3] <= Length[p] + 1], {n, 1, z}] (* A239469 *)
t4 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 4] <= Length[p] + 1], {n, 1, z}] (* A239470 *)
t5 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 5] <= Length[p] + 1], {n, 1, z}] (* A239472 *)

A239470 Number of 4-separable partitions of n; see Comments.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 2, 2, 3, 5, 6, 7, 8, 11, 13, 17, 19, 23, 27, 34, 40, 47, 55, 66, 77, 92, 106, 125, 145, 171, 198, 231, 266, 310, 358, 416, 477, 552, 633, 731, 838, 963, 1100, 1263, 1442, 1651, 1880, 2147, 2442, 2785, 3163, 3597, 4078, 4631, 5244, 5946

Offset: 1

Clark Kimberling, Mar 20 2014

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.

			(4,0)-separable partitions of 7: 241;
(4,1)-separable partitions of 7: 43;
(4,2)-separable partitions of 7: (none);
4-separable partitions of 7: 241, 43, so that a(7) = 2.

Cf. A239467, A239468, A239469, A239471.

z = 55; t1 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 1] <= Length[p] + 1], {n, 1, z}] (* A239467 *)
t2 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 2] <= Length[p] + 1], {n, 1, z}] (* A239468 *)
t3 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 3] <= Length[p] + 1], {n, 1, z}] (* A239469 *)
t4 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 4] <= Length[p] + 1], {n, 1, z}] (* A239470 *)
t5 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 5] <= Length[p] + 1], {n, 1, z}] (* A239472 *)

A239471 Number of 5-separable partitions of n; see Comments.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 2, 3, 2, 4, 5, 7, 8, 9, 11, 13, 16, 20, 23, 27, 31, 37, 43, 52, 59, 70, 80, 93, 108, 126, 144, 167, 191, 221, 253, 292, 332, 382, 435, 498, 567, 649, 736, 839, 951, 1082, 1226, 1393, 1573, 1784, 2013, 2277, 2568, 2902, 3266, 3683, 4141

Offset: 1

Clark Kimberling, Mar 20 2014

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.

			(5,0)-separable partitions of 7: 151
(5,1)-separable partitions of 7: 52
(5,2)-separable partitions of 7: (none)
5-separable partitions of 7: 151, 52, so that a(7) = 2.

Cf. A239467, A239468, A239469, A239470.

z = 55; t1 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 1] <= Length[p] + 1], {n, 1, z}] (* A239467 *)
t2 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 2] <= Length[p] + 1], {n, 1, z}] (* A239468 *)
t3 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 3] <= Length[p] + 1], {n, 1, z}] (* A239469 *)
t4 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 4] <= Length[p] + 1], {n, 1, z}] (* A239470 *)
t5 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 5] <= Length[p] + 1], {n, 1, z}] (* A239472 *)

A239483 Number of (3,0)-separable partitions of n; see Comments.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 3, 4, 5, 7, 8, 10, 12, 16, 18, 22, 26, 33, 38, 45, 53, 65, 75, 89, 103, 124, 143, 168, 195, 230, 265, 309, 357, 418, 479, 556, 639, 742, 850, 979, 1122, 1294, 1478, 1696, 1935, 2220, 2528, 2889, 3287, 3752, 4261, 4850, 5502, 6257, 7084

Offset: 4

Clark Kimberling, Mar 20 2014

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.

			The (3,0)-separable partitions of 11 are 731, 632, 434, 23231, so that a(11) = 4.

Cf. A239467, A165652, A239482, A239484, A239485.

z = 65; -1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 1] == Length[p] - 1], {n, 2, z}]  (* A165652 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2] == Length[p] - 1], {n, 3, z}]  (* A239482 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 3] == Length[p] - 1], {n, 4, z}]  (* A239483 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 4] == Length[p] - 1], {n, 5, z}]  (* A239484 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 5] == Length[p] - 1], {n, 6, z}] (* A239485 *)

A239484 Number of (4,0)-separable partitions of n; see Comments.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 19, 22, 26, 31, 36, 42, 51, 58, 68, 79, 92, 107, 125, 143, 165, 191, 221, 253, 293, 333, 383, 440, 503, 574, 657, 747, 853, 971, 1105, 1253, 1427, 1616, 1833, 2076, 2349, 2655, 3006, 3389, 3826, 4313, 4861

Offset: 5

Clark Kimberling, Mar 20 2014

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.

			The (4,0)-separable partitions of 12 are 741, 642, 543, 24141, so that a(12) = 4.

Cf. A239467, A165652, A239482, A239483, A239485.

z = 65; -1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 1] == Length[p] - 1], {n, 2, z}]  (* A165652 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2] == Length[p] - 1], {n, 3, z}]  (* A239482 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 3] == Length[p] - 1], {n, 4, z}]  (* A239483 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 4] == Length[p] - 1], {n, 5, z}]  (* A239484 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 5] == Length[p] - 1], {n, 6, z}] (* A239485 *)

A239485 Number of (5,0)-separable partitions of n; see Comments.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 4, 4, 6, 7, 8, 9, 12, 13, 16, 19, 22, 25, 31, 34, 41, 47, 54, 62, 74, 82, 96, 110, 126, 143, 167, 187, 216, 245, 279, 316, 364, 408, 466, 527, 597, 673, 767, 860, 976, 1098, 1238, 1391, 1574, 1761, 1986, 2228, 2502, 2801, 3150, 3518

Offset: 6

Clark Kimberling, Mar 20 2014

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.

			The (5,0)-separable partitions of 13 are 751, 652, 454, 15151, so that a(13) = 4.

Cf. A239467, A239481, A165652, A239483, A239484.

z = 65; -1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 1] == Length[p] - 1], {n, 2, z}]  (* A165652 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2] == Length[p] - 1], {n, 3, z}]  (* A239482 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 3] == Length[p] - 1], {n, 4, z}]  (* A239483 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 4] == Length[p] - 1], {n, 5, z}]  (* A239484 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 5] == Length[p] - 1], {n, 6, z}] (* A239485 *)

cp's OEIS Frontend

A239482 Number of (2,0)-separable partitions of n; see Comments.

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A239468 Number of 2-separable partitions of n; see Comments.

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A239469 Number of 3-separable partitions of n; see Comments.

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A239470 Number of 4-separable partitions of n; see Comments.

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A239471 Number of 5-separable partitions of n; see Comments.

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A239483 Number of (3,0)-separable partitions of n; see Comments.

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A239484 Number of (4,0)-separable partitions of n; see Comments.

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A239485 Number of (5,0)-separable partitions of n; see Comments.

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