A241086 -id:A241086

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A241089

Number of partitions p of n into distinct parts such that max(p) > 2*(number of parts of p).

+10
8

0, 0, 0, 1, 1, 1, 2, 3, 4, 5, 6, 8, 10, 13, 16, 20, 24, 29, 35, 42, 50, 61, 72, 85, 101, 118, 138, 161, 188, 218, 254, 293, 339, 391, 450, 515, 591, 675, 771, 878, 999, 1135, 1289, 1460, 1652, 1868, 2108, 2376, 2676, 3009, 3379, 3793, 4250, 4760, 5325, 5952 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,7`
	LINKS	`Table of n, a(n) for n=0..55.`
	EXAMPLE	`a(9) counts these 5 partitions: 9, 81, 72, 63, 54.`
	MATHEMATICA	`z = 40; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];` `Table[Count[f[n], p_ /; Max[p] < 2Length[p]], {n, 0, z}] ( A241085 )` `Table[Count[f[n], p_ /; Max[p] <= 2Length[p]], {n, 0, z}] (* A241086 )` `Table[Count[f[n], p_ /; Max[p] == 2Length[p]], {n, 0, z}] (* A241087 )` `Table[Count[f[n], p_ /; Max[p] >= 2Length[p]], {n, 0, z}] (* A241088 )` `Table[Count[f[n], p_ /; Max[p] > 2Length[p]], {n, 0, z}] (* A241089 *)`
	CROSSREFS	`Cf. A241085, A241086, A241087, A241088.`
	KEYWORD	`nonn,easy`
	AUTHOR	`Clark Kimberling, Apr 17 2014`
	STATUS	`approved`

A241087

Number of partitions p of n into distinct parts such that max(p) = 2*(number of parts of p).

+10
7

0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 6, 5, 6, 6, 7, 7, 9, 10, 12, 13, 15, 16, 18, 19, 20, 23, 25, 28, 30, 35, 38, 43, 46, 51, 55, 61, 64, 72, 76, 84, 91, 101, 109, 120, 130, 142, 155, 168, 181, 196, 212, 228, 248, 266, 288, 311, 337 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,12`
	LINKS	`Table of n, a(n) for n=0..66.`
	EXAMPLE	`a(15) counts these 2 partitions: 8421, 654.`
	MATHEMATICA	`z = 40; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];` `Table[Count[f[n], p_ /; Max[p] < 2Length[p]], {n, 0, z}] ( A241085 )` `Table[Count[f[n], p_ /; Max[p] <= 2Length[p]], {n, 0, z}] (* A241086 )` `Table[Count[f[n], p_ /; Max[p] == 2Length[p]], {n, 0, z}] (* A241087 )` `Table[Count[f[n], p_ /; Max[p] >= 2Length[p]], {n, 0, z}] (* A241088 )` `Table[Count[f[n], p_ /; Max[p] > 2Length[p]], {n, 0, z}] (* A241089 *)`
	CROSSREFS	`Cf. A241085, A241086, A241088, A241089.`
	KEYWORD	`nonn,easy`
	AUTHOR	`Clark Kimberling, Apr 17 2014`
	STATUS	`approved`

A241085

Number of partitions p of n into distinct parts such that max(p) < 2*(number of parts of p).

+10
5

0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 2, 3, 3, 4, 5, 6, 6, 7, 8, 8, 10, 11, 13, 14, 17, 18, 21, 22, 25, 27, 31, 33, 38, 42, 47, 52, 57, 63, 69, 76, 82, 91, 99, 109, 119, 132, 142, 158, 171, 188, 203, 223, 240, 263, 284, 309, 334, 364, 393, 428, 463, 501, 543, 588 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,9`
	LINKS	`Table of n, a(n) for n=0..64.`
	EXAMPLE	`a(15) counts these 5 partitions: 7521, 7431, 6531, 6432, 54321.`
	MATHEMATICA	`z = 40; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];` `Table[Count[f[n], p_ /; Max[p] < 2Length[p]], {n, 0, z}] ( A241085 )` `Table[Count[f[n], p_ /; Max[p] <= 2Length[p]], {n, 0, z}] (* A241086 )` `Table[Count[f[n], p_ /; Max[p] == 2Length[p]], {n, 0, z}] (* A241087 )` `Table[Count[f[n], p_ /; Max[p] >= 2Length[p]], {n, 0, z}] (* A241088 )` `Table[Count[f[n], p_ /; Max[p] > 2Length[p]], {n, 0, z}] (* A241089 *)`
	CROSSREFS	`Cf. A241086, A241087, A241088, A241089.`
	KEYWORD	`nonn,easy`
	AUTHOR	`Clark Kimberling, Apr 17 2014`
	STATUS	`approved`

A241088

Number of partitions p of n into distinct parts such that max(p) >= 2*(number of parts of p).

+10
5

0, 0, 1, 1, 1, 2, 3, 4, 4, 6, 7, 10, 12, 15, 18, 22, 26, 32, 39, 46, 56, 66, 78, 91, 108, 125, 147, 171, 200, 231, 269, 309, 357, 410, 470, 538, 616, 703, 801, 913, 1037, 1178, 1335, 1511, 1707, 1929, 2172, 2448, 2752, 3093, 3470, 3894, 4359, 4880, 5455 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,6`
	LINKS	`Table of n, a(n) for n=0..54.`
	EXAMPLE	`a(9) counts these 6 partitions: 9, 81, 72, 63, 621, 54.`
	MATHEMATICA	`z = 40; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];` `Table[Count[f[n], p_ /; Max[p] < 2Length[p]], {n, 0, z}] ( A241085 )` `Table[Count[f[n], p_ /; Max[p] <= 2Length[p]], {n, 0, z}] (* A241086 )` `Table[Count[f[n], p_ /; Max[p] == 2Length[p]], {n, 0, z}] (* A241087 )` `Table[Count[f[n], p_ /; Max[p] >= 2Length[p]], {n, 0, z}] (* A241088 )` `Table[Count[f[n], p_ /; Max[p] > 2Length[p]], {n, 0, z}] (* A241089 *)`
	CROSSREFS	`Cf. A241085, A241086, A241087, A241089.`
	KEYWORD	`nonn,easy`
	AUTHOR	`Clark Kimberling, Apr 17 2014`
	STATUS	`approved`

A241091

Number of partitions p of n into distinct parts such that max(p) <= 1 + 2*(number of parts of p).

+10
3

0, 1, 1, 2, 1, 2, 3, 3, 3, 4, 5, 5, 7, 7, 9, 10, 11, 12, 15, 16, 19, 22, 24, 27, 30, 34, 37, 43, 47, 53, 59, 66, 72, 82, 88, 99, 109, 120, 131, 146, 160, 176, 194, 212, 233, 256, 279, 304, 334, 362, 396, 431, 471, 510, 558, 604, 659, 714, 776, 839, 913, 985 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,4`
	LINKS	`Table of n, a(n) for n=0..61.`
	FORMULA	`a(n) = A241086(n) + A241092(n) for n >= 0.`
	EXAMPLE	`a(12) counts these 5 partitions: 741, 732, 651, 642, 6321, 543, 5421.`
	MATHEMATICA	`z = 30; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];` `Table[Count[f[n], p_ /; Max[p] < 1 + 2Length[p]], {n, 0, z}] (A241086)` `Table[Count[f[n], p_ /; Max[p] <= 1 + 2Length[p]], {n, 0, z}](A241091)` `Table[Count[f[n], p_ /; Max[p] == 1 + 2Length[p]], {n, 0, z}](A241092)` `Table[Count[f[n], p_ /; Max[p] >= 1 + 2Length[p]], {n, 0, z}](A241089)` `Table[Count[f[n], p_ /; Max[p] > 1 + 2Length[p]], {n, 0, z}] (A241093*)`
	CROSSREFS	`Cf. A241086, A241092, A241093.`
	KEYWORD	`nonn,easy`
	AUTHOR	`Clark Kimberling, Apr 18 2014`
	STATUS	`approved`

A241092

Number of partitions p of n into distinct parts such that max(p) = 1 + 2*(number of parts of p).

+10
3

0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 7, 7, 8, 9, 10, 10, 12, 13, 15, 17, 19, 21, 25, 26, 29, 32, 35, 38, 42, 46, 51, 57, 62, 69, 76, 83, 90, 100, 107, 117, 127, 139, 150, 165, 178, 195, 212, 231, 250, 273, 294, 319, 346, 373, 402 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,13`
	LINKS	`Table of n, a(n) for n=0..65.`
	FORMULA	`a(n) + A241086(n) + A241093(n) = A000009(n) for n >= 1.` `a(n) = A241091(n) - A241086(n) for n >= 0.`
	EXAMPLE	`a(12) counts these 5 partitions: 741, 732, 651, 642, 6321, 543, 5421.`
	MATHEMATICA	`z = 30; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];` `Table[Count[f[n], p_ /; Max[p] < 1 + 2Length[p]], {n, 0, z}] (A241086)` `Table[Count[f[n], p_ /; Max[p] <= 1 + 2Length[p]], {n, 0, z}](A241091)` `Table[Count[f[n], p_ /; Max[p] == 1 + 2Length[p]], {n, 0, z}](A241092)` `Table[Count[f[n], p_ /; Max[p] >= 1 + 2Length[p]], {n, 0, z}](A241089)` `Table[Count[f[n], p_ /; Max[p] > 1 + 2Length[p]], {n, 0, z}] (A241093*)`
	CROSSREFS	`Cf. A241086, A241091, A241093, A000009.`
	KEYWORD	`nonn,easy`
	AUTHOR	`Clark Kimberling, Apr 18 2014`
	STATUS	`approved`

A241093

Number of partitions p of n into distinct parts such that max(p) > 1 + 2*(number of parts of p).

+10
3

0, 0, 0, 0, 1, 1, 1, 2, 3, 4, 5, 7, 8, 11, 13, 17, 21, 26, 31, 38, 45, 54, 65, 77, 92, 108, 128, 149, 175, 203, 237, 274, 318, 366, 424, 486, 559, 640, 733, 836, 953, 1084, 1232, 1398, 1583, 1792, 2025, 2286, 2576, 2902, 3262, 3666, 4111, 4610, 5160, 5774 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,8`
	LINKS	`Table of n, a(n) for n=0..55.`
	FORMULA	`a(n) + A241086(n) + A241093(n) = A000009(n) for n >= 1.`
	EXAMPLE	`a(12) counts these 8 partitions: {12}, {11,1}, {10,2}, {9,3}, 9,2,1}, {8,4}, {8,3,1}, {7,5}.`
	MATHEMATICA	`z = 30; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];` `Table[Count[f[n], p_ /; Max[p] < 1 + 2Length[p]], {n, 0, z}] (A241086)` `Table[Count[f[n], p_ /; Max[p] <= 1 + 2Length[p]], {n, 0, z}](A241091)` `Table[Count[f[n], p_ /; Max[p] == 1 + 2Length[p]], {n, 0, z}](A241092)` `Table[Count[f[n], p_ /; Max[p] >= 1 + 2Length[p]], {n, 0, z}](A241089)` `Table[Count[f[n], p_ /; Max[p] > 1 + 2Length[p]], {n, 0, z}] (A241093*)`
	CROSSREFS	`Cf. A241086, A241091, A241092, A000009.`
	KEYWORD	`nonn,easy`
	AUTHOR	`Clark Kimberling, Apr 18 2014`
	STATUS	`approved`

A363221

Number of strict integer partitions of n such that (length) * (maximum) <= 2n.

+10
0

1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 15, 19, 23, 26, 29, 37, 39, 49, 55, 62, 71, 84, 93, 108, 118, 141, 149, 188, 193, 217, 257, 279, 318, 369, 376, 441, 495, 572, 587, 692, 760, 811, 960, 1046, 1065, 1307, 1387, 1550, 1703, 1796, 2041, 2295, 2456, 2753, 3014 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,3`
	COMMENTS	`Also strict partitions such that (maximum) <= 2*(mean).` `These are strict partitions whose complement (see A361851) has size <= n.`
	LINKS	`Table of n, a(n) for n=1..57.`
	EXAMPLE	`The partition y = (4,3,1) has length 3 and maximum 4, and 34 <= 28, so y is counted under a(8). The complement of y has size 4, which is less than or equal to n = 8.`
	MATHEMATICA	`Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Max@@#<=2*Mean[#]&]], {n, 30}]`
	CROSSREFS	`The equal case for median is A361850, non-strict A361849 (ranks A361856).` `The non-strict version is A361851, A361848 for median.` `The equal case is A361854, non-strict A361853 (ranks A361855).` `A000041 counts integer partitions, strict A000009.` `A008284 counts partitions by length, A058398 by mean.` `A051293 counts subsets with integer mean.` `A067538 counts partitions with integer mean.` `Cf. A111907, A237984, A240219, A241061, A241086, A324521, A324562, A349156, A360068, A360241, A361394, A361852, A361906.`
	KEYWORD	`nonn`
	AUTHOR	`Gus Wiseman, May 23 2023`
	STATUS	`approved`

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Last modified August 29 21:13 EDT 2024. Contains 375518 sequences. (Running on oeis4.)