A382857
Number of ways to permute the prime indices of n so that the run-lengths are all equal.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 0, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 2, 4, 1, 2, 2, 0, 1, 6, 1, 1, 1, 2, 1, 0, 1, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 6, 1, 2, 1, 1, 2, 6, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 2, 6, 1, 0, 1, 2, 1, 6, 2, 2
Offset: 0
The prime indices of 216 are {1,1,1,2,2,2} and we have permutations:
(1,1,1,2,2,2)
(1,2,1,2,1,2)
(2,1,2,1,2,1)
(2,2,2,1,1,1)
so a(216) = 4.
The prime indices of 25920 are {1,1,1,1,1,1,2,2,2,2,3} and we have permutations:
(1,2,1,2,1,2,1,2,1,3,1)
(1,2,1,2,1,2,1,3,1,2,1)
(1,2,1,2,1,3,1,2,1,2,1)
(1,2,1,3,1,2,1,2,1,2,1)
(1,3,1,2,1,2,1,2,1,2,1)
so a(25920) = 5.
For distinct instead of equal run-lengths we have
A382771.
For run-sums instead of run-lengths we have
A382877, distinct
A382876.
Positions of first appearances are
A382878.
Positions of terms > 1 are
A383089.
A003963 gives product of prime indices.
A005811 counts runs in binary expansion.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A164707 lists numbers whose binary expansion has all equal run-lengths, distinct
A328592.
A353744 ranks compositions with equal run-lengths, counted by
A329738.
Cf.
A000720,
A000961,
A001221,
A001222,
A003242,
A008480,
A047966,
A238130,
A238279,
A351201,
A351293,
A351295.
-
Table[Length[Select[Permutations[Join@@ConstantArray@@@FactorInteger[n]], SameQ@@Length/@Split[#]&]],{n,0,100}]
A383015
Numbers whose prime indices have more than one permutation with all equal run-sums.
Original entry on oeis.org
12, 40, 63, 112, 144, 325, 351, 352, 675, 832, 931, 1008, 1539, 1600, 1728, 2176, 2875, 3509, 3969, 4864, 6253, 7047, 7056, 8775, 9072, 11776, 12427, 12544, 12691, 16128, 19133, 20736, 20800, 22464, 23125, 26973, 29403, 29696, 32269, 43200, 49392, 57967, 59711
Offset: 1
The terms together with their prime indices begin:
12: {1,1,2}
40: {1,1,1,3}
63: {2,2,4}
112: {1,1,1,1,4}
144: {1,1,1,1,2,2}
325: {3,3,6}
351: {2,2,2,6}
352: {1,1,1,1,1,5}
675: {2,2,2,3,3}
832: {1,1,1,1,1,1,6}
931: {4,4,8}
1008: {1,1,1,1,2,2,4}
1539: {2,2,2,2,8}
1600: {1,1,1,1,1,1,3,3}
1728: {1,1,1,1,1,1,2,2,2}
The complement for run-lengths instead of sums is
A383091, counted by
A383092Partitions of this type are counted by
A383097.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A353847 gives composition run-sum transformation, for partitions
A353832.
A353932 lists run-sums of standard compositions.
Cf.
A000720,
A000961,
A001221,
A001222,
A329738,
A353833,
A354584,
A381636,
A381871,
A382857,
A382876,
A382879.
A383097
Number of integer partitions of n having more than one permutation with all equal run-sums.
Original entry on oeis.org
0, 0, 0, 0, 1, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 9, 0, 7, 0, 12, 0, 1, 0, 38, 0, 1, 1, 18, 0, 38, 0, 32, 0, 1, 0, 90, 0, 1, 0, 71, 0, 78, 0, 33, 10, 1, 0, 228, 0, 31, 0, 42, 0, 156, 0, 123, 0, 1, 0, 447, 0, 1, 16, 146, 0, 222, 0, 63, 0, 102, 0, 811, 0, 1, 29, 75, 0, 334, 0
Offset: 0
The a(27) = 1 partition is: (9,3,3,3,1,1,1,1,1,1,1,1,1).
The a(4) = 1 through a(16) = 9 partitions (empty columns not shown):
(211) (3111) (422) (511111) (633) (71111111) (844)
(41111) (6222) (82222)
(221111) (33222) (442222)
(4221111) (44221111)
(6111111) (422221111)
(33111111) (811111111)
(222111111) (4411111111)
(42211111111)
(222211111111)
These partitions are ranked by
A383015, positions of terms > 1 in
A382877.
For any positive number of permutations we have
A383098, ranks
A383110.
Counting and ranking partitions by run-lengths and run-sums:
A382876 counts permutations of prime indices with distinct run-sums, zeros
A381636.
-
Table[Length[Select[IntegerPartitions[n],Length[Select[Permutations[#],SameQ@@Total/@Split[#]&]]>1&]],{n,0,15}]
A383099
Numbers whose prime indices have exactly one permutation with all equal run-sums.
Original entry on oeis.org
1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 41, 43, 47, 48, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193
Offset: 1
The terms together with their prime indices begin:
1: {}
2: {1}
3: {2}
4: {1,1}
5: {3}
7: {4}
8: {1,1,1}
9: {2,2}
11: {5}
13: {6}
16: {1,1,1,1}
17: {7}
19: {8}
23: {9}
25: {3,3}
27: {2,2,2}
29: {10}
31: {11}
32: {1,1,1,1,1}
36: {1,1,2,2}
37: {12}
41: {13}
For distinct instead of equal run-sums we have
A000961, counted by
A000005.
These are the positions of 1 in
A382877.
For more than one choice we have
A383015.
Partitions of this type are counted by
A383095.
For run-lengths instead of sums we have
A383112 = positions of 1 in
A382857.
A383094
Number of integer partitions of n having exactly one permutation with all equal run-lengths.
Original entry on oeis.org
1, 1, 2, 2, 4, 4, 5, 6, 9, 7, 11, 10, 13, 12, 17, 14, 21, 16, 21, 18, 27, 22, 29, 22, 34, 25, 35, 28, 41, 28, 43, 30, 48, 38, 47, 38, 55, 36, 53, 46, 64, 40, 67, 42, 69, 54, 65, 46, 84, 51, 75, 62, 83, 52, 86, 62, 94, 70, 83, 58, 111, 60, 89, 80, 106, 74, 115, 66, 111
Offset: 0
The partition (222211) has exactly one permutation with all equal run-lengths: (221122), so is counted under a(10).
The a(1) = 1 through a(8) = 9 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (221) (33) (322) (44)
(211) (311) (222) (331) (332)
(1111) (11111) (411) (511) (422)
(111111) (22111) (611)
(1111111) (2222)
(22211)
(221111)
(11111111)
Partitions of this type are ranked by
A383112 = positions of 1 in
A382857.
-
Table[Length[Select[IntegerPartitions[n], Length[Select[Permutations[#], SameQ@@Length/@Split[#]&]]==1&]],{n,0,20}]
A383090
Number of integer partitions of n having more than one permutation with all equal run-lengths.
Original entry on oeis.org
0, 0, 0, 1, 1, 2, 4, 5, 9, 14, 20, 28, 43, 55, 77, 107, 141, 183, 244, 312, 411, 521, 664, 837, 1069, 1328, 1667, 2069, 2578, 3166, 3929, 4791, 5895, 7168, 8749, 10594, 12883, 15500, 18741, 22493, 27069, 32334, 38760, 46133, 55065, 65367, 77686, 91905, 108927, 128431, 151674
Offset: 0
The partition (3322221) has 3 permutations with all equal run-lengths: (2323212), (2321232), (2123232), so is counted under a(15).
The partition (3322111111) has 2 permutations with all equal run-lengths: (1133112211), (1122113311), so is counted under a(16).
The a(3) = 1 through a(9) = 14 partitions:
(21) (31) (32) (42) (43) (53) (54)
(41) (51) (52) (62) (63)
(321) (61) (71) (72)
(2211) (421) (431) (81)
(3211) (521) (432)
(3221) (531)
(3311) (621)
(4211) (3321)
(32111) (4221)
(4311)
(5211)
(32211)
(42111)
(222111)
Partitions of this type are ranked by
A383089 = positions of terms > 1 in
A382857.
For distinct instead of equal run-lengths we have
A383111, ranks
A383113.
-
Table[Length[Select[IntegerPartitions[n], Length[Select[Permutations[#], SameQ@@Length/@Split[#]&]]>1&]],{n,0,15}]
A383112
Numbers whose multiset of prime indices has exactly one permutation with all equal run-lengths.
Original entry on oeis.org
1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 25, 27, 28, 29, 31, 32, 37, 41, 43, 44, 45, 47, 49, 50, 52, 53, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 81, 83, 89, 92, 97, 98, 99, 101, 103, 107, 108, 109, 113, 116, 117, 121, 124, 125, 127
Offset: 1
The prime indices of 144 are {1,1,1,1,2,2}, of which the only permutation with all equal run-lengths is (1,1,2,2,1,1), so 144 is in the sequence.
The terms together with their prime indices begin:
1: {}
2: {1}
3: {2}
4: {1,1}
5: {3}
7: {4}
8: {1,1,1}
9: {2,2}
11: {5}
12: {1,1,2}
13: {6}
16: {1,1,1,1}
17: {7}
18: {1,2,2}
19: {8}
20: {1,1,3}
23: {9}
25: {3,3}
27: {2,2,2}
28: {1,1,4}
29: {10}
31: {11}
32: {1,1,1,1,1}
Partitions of this type are counted by
A383094.
Cf.
A000961,
A001221,
A001222,
A048767,
A351294,
A351295,
A353833,
A381434,
A381540,
A382877,
A383100.
A383091
Numbers whose prime indices have at most one permutation with all equal run-lengths.
Original entry on oeis.org
1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109
Offset: 1
The prime indices of 144 are {1,1,1,1,2,2}, with just one permutation with all equal run-lengths (1,1,2,2,1,1), so 144 is in the sequence.
The prime indices of 240 are {1,1,1,1,2,3}, which have no permutation with all equal run-lengths, so 240 is in the sequence.
The terms together with their prime indices begin:
1: {}
2: {1}
3: {2}
4: {1,1}
5: {3}
7: {4}
8: {1,1,1}
9: {2,2}
11: {5}
12: {1,1,2}
13: {6}
16: {1,1,1,1}
17: {7}
18: {1,2,2}
19: {8}
20: {1,1,3}
23: {9}
24: {1,1,1,2}
The complement for run-sums instead of lengths is
A383015, counted by
A383097.
A047966 counts partitions with equal run-lengths, compositions
A329738.
Cf.
A000720,
A000961,
A001221,
A001222,
A048767,
A351294,
A353744,
A353833,
A381435,
A382771,
A382877,
A383113.
A383092
Number of integer partitions of n having at most one permutation with all equal run-lengths.
Original entry on oeis.org
1, 1, 2, 2, 4, 5, 7, 10, 13, 16, 22, 28, 34, 46, 58, 69, 90, 114, 141, 178, 216, 271, 338, 418, 506, 630, 769, 941, 1140, 1399, 1675, 2051, 2454, 2975, 3561, 4289, 5094, 6137, 7274, 8692, 10269, 12249, 14414, 17128, 20110, 23767, 27872, 32849, 38346, 45094, 52552, 61533
Offset: 0
The partition (222211) has 1 permutation with all equal run-lengths: (221122), so is counted under a(10).
The partition (33211111) has no permutation with all equal run-lengths, so is counted under a(13).
The a(1) = 1 through a(7) = 10 partitions:
(1) (2) (3) (4) (5) (6) (7)
(11) (111) (22) (221) (33) (322)
(211) (311) (222) (331)
(1111) (2111) (411) (511)
(11111) (3111) (2221)
(21111) (4111)
(111111) (22111)
(31111)
(211111)
(1111111)
Partitions of this type are ranked by
A383091 = positions of terms <= 1 in
A382857.
-
Table[Length[Select[IntegerPartitions[n],Length[Select[Permutations[#],SameQ@@Length/@Split[#]&]]<=1&]],{n,0,15}]
A383113
Numbers whose prime indices have more than one permutation with all distinct run-lengths.
Original entry on oeis.org
12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 63, 68, 72, 75, 76, 80, 88, 92, 96, 98, 99, 104, 108, 112, 116, 117, 124, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 171, 172, 175, 176, 184, 188, 189, 192, 200, 207, 208, 212, 216, 224, 232, 236, 242
Offset: 1
The prime indices of 360 are {1,1,1,2,2,3}, with six permutations with all distinct run-lengths:
(1,1,1,2,2,3)
(1,1,1,3,2,2)
(2,2,1,1,1,3)
(2,2,3,1,1,1)
(3,1,1,1,2,2)
(3,2,2,1,1,1)
so 360 is in the sequence.
The terms together with their prime indices begin:
12: {1,1,2}
18: {1,2,2}
20: {1,1,3}
24: {1,1,1,2}
28: {1,1,4}
40: {1,1,1,3}
44: {1,1,5}
45: {2,2,3}
48: {1,1,1,1,2}
50: {1,3,3}
52: {1,1,6}
54: {1,2,2,2}
56: {1,1,1,4}
63: {2,2,4}
68: {1,1,7}
72: {1,1,1,2,2}
75: {2,3,3}
76: {1,1,8}
80: {1,1,1,1,3}
For equal run-lengths we have
A383089, positions of terms > 1 in
A382857.
Partitions of this type are counted by
A383111.
A044813 lists numbers whose binary expansion has distinct run-lengths.
Cf.
A000720,
A001221,
A001222,
A047966,
A048767,
A351013,
A351202,
A381435,
A382876,
A383090,
A383091,
A383092,
A383112.
Showing 1-10 of 11 results.
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