A382879
Positions of 0 in A382857 (permutations of prime indices with equal run-lengths).
Original entry on oeis.org
24, 40, 48, 54, 56, 80, 88, 96, 104, 112, 135, 136, 152, 160, 162, 176, 184, 189, 192, 208, 224, 232, 240, 248, 250, 272, 288, 296, 297, 304, 320, 328, 336, 344, 351, 352, 368, 375, 376, 384, 405, 416, 424, 448, 459, 464, 472, 480, 486, 488, 496, 513, 528, 536
Offset: 1
The terms together with their prime indices begin:
24: {1,1,1,2}
40: {1,1,1,3}
48: {1,1,1,1,2}
54: {1,2,2,2}
56: {1,1,1,4}
80: {1,1,1,1,3}
88: {1,1,1,5}
96: {1,1,1,1,1,2}
104: {1,1,1,6}
112: {1,1,1,1,4}
135: {2,2,2,3}
136: {1,1,1,7}
152: {1,1,1,8}
160: {1,1,1,1,1,3}
For distinct instead of equal the complement is
A351294, counted by
A239455.
For prime signature instead of prime indices we have
A382914.
Partitions of this type are counted by
A382915.
The complement is counted by
A383013.
A005811 counts runs in binary expansion.
A297770 counts distinct runs in binary expansion.
A164707 lists numbers whose binary form has equal runs of ones, distinct
A328592.
A351290
Numbers k such that the k-th composition in standard order has all distinct runs.
Original entry on oeis.org
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 50, 51, 52, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 78
Offset: 1
The terms together with their binary expansions and corresponding compositions begin:
0: 0 ()
1: 1 (1)
2: 10 (2)
3: 11 (1,1)
4: 100 (3)
5: 101 (2,1)
6: 110 (1,2)
7: 111 (1,1,1)
8: 1000 (4)
9: 1001 (3,1)
10: 1010 (2,2)
11: 1011 (2,1,1)
12: 1100 (1,3)
14: 1110 (1,1,2)
15: 1111 (1,1,1,1)
The version for Heinz numbers and prime multiplicities is
A130091.
The version for run-lengths instead of runs is
A329739.
These compositions are counted by
A351013.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A085207 represents concatenation of standard compositions, reverse
A085208.
A351204 counts partitions where every permutation has all distinct runs.
Counting words with all distinct runs:
-
A351202 = permutations of prime factors.
Selected statistics of standard compositions:
- Number of distinct parts is
A334028.
Selected classes of standard compositions:
- Constant compositions are
A272919.
Cf.
A098859,
A106356,
A113835,
A116608,
A238279,
A242882,
A318928,
A325545,
A328592,
A329745,
A350952,
A351015,
A351201.
-
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],UnsameQ@@Split[stc[#]]&]
A351291
Numbers k such that the k-th composition in standard order does not have all distinct runs.
Original entry on oeis.org
13, 22, 25, 45, 46, 49, 53, 54, 59, 76, 77, 82, 89, 91, 93, 94, 97, 101, 102, 105, 108, 109, 110, 115, 118, 141, 148, 150, 153, 156, 162, 165, 166, 173, 177, 178, 180, 181, 182, 183, 187, 189, 190, 193, 197, 198, 201, 204, 205, 209, 210, 213, 214, 216, 217
Offset: 1
The terms together with their binary expansions and corresponding compositions begin:
13: 1101 (1,2,1)
22: 10110 (2,1,2)
25: 11001 (1,3,1)
45: 101101 (2,1,2,1)
46: 101110 (2,1,1,2)
49: 110001 (1,4,1)
53: 110101 (1,2,2,1)
54: 110110 (1,2,1,2)
59: 111011 (1,1,2,1,1)
76: 1001100 (3,1,3)
77: 1001101 (3,1,2,1)
82: 1010010 (2,3,2)
89: 1011001 (2,1,3,1)
91: 1011011 (2,1,2,1,1)
93: 1011101 (2,1,1,2,1)
94: 1011110 (2,1,1,1,2)
The version for Heinz numbers of partitions is
A130092, complement
A130091.
Normal multisets with a permutation of this type appear to be
A283353.
Partitions w/o permutations of this type are
A351204, complement
A351203.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has all distinct run-lengths.
A085207 represents concatenation of standard compositions, reverse
A085208.
Counting words with all distinct runs:
-
A351202 = permutations of prime factors.
Selected statistics of standard compositions (
A066099, reverse
A228351):
- Number of distinct parts is
A334028.
Selected classes of standard compositions:
- Constant compositions are
A272919.
Cf.
A098859,
A106356,
A113835,
A116608,
A238279,
A242882,
A318928,
A325545,
A328592,
A329745,
A350952,
A351015,
A351201.
-
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!UnsameQ@@Split[stc[#]]&]
A383013
Number of integer partitions of n having a permutation with all equal run-lengths.
Original entry on oeis.org
1, 1, 2, 3, 5, 6, 9, 11, 18, 21, 31, 38, 56, 67, 94, 121, 162, 199, 265, 330, 438, 543, 693, 859, 1103, 1353, 1702, 2097, 2619, 3194, 3972, 4821, 5943, 7206, 8796, 10632, 12938, 15536, 18794, 22539, 27133, 32374, 38827, 46175, 55134, 65421, 77751, 91951, 109011, 128482
Offset: 0
The partition (2,2,1,1,1,1) has permutation (1,1,2,2,1,1) with equal run-lengths (2,2,2) so is counted under a(8).
The a(1) = 1 through a(8) = 18 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (43) (44)
(111) (31) (41) (42) (52) (53)
(211) (221) (51) (61) (62)
(1111) (311) (222) (322) (71)
(11111) (321) (331) (332)
(411) (421) (422)
(2211) (511) (431)
(111111) (3211) (521)
(22111) (611)
(1111111) (2222)
(3221)
(3311)
(4211)
(22211)
(32111)
(221111)
(11111111)
For distinct instead of equal run-lengths we have
A239455, ranked by
A351294.
The complement for distinct run-lengths is
A351293, ranked by
A351295.
A382857 counts permutations of prime indices with equal run-lengths, firsts
A382878.
-
Table[Length[Select[IntegerPartitions[n],Select[Permutations[#], SameQ@@Length/@Split[#]&]!={}&]],{n,0,15}]
A351204
Number of integer partitions of n such that every permutation has all distinct runs.
Original entry on oeis.org
1, 1, 2, 3, 4, 5, 8, 9, 11, 14, 18, 20, 25, 28, 34, 41, 47, 53, 64, 72, 84, 98, 113, 128, 148, 169, 194, 223, 255, 289, 333, 377, 428, 488, 554, 629, 715, 807, 913, 1033, 1166, 1313, 1483, 1667, 1875, 2111, 2369, 2655, 2977, 3332, 3729, 4170, 4657, 5195, 5797, 6459
Offset: 0
The a(1) = 1 through a(8) = 11 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (43) (44)
(111) (31) (41) (42) (52) (53)
(1111) (2111) (51) (61) (62)
(11111) (222) (421) (71)
(321) (2221) (431)
(3111) (4111) (521)
(111111) (211111) (2222)
(1111111) (5111)
(311111)
(11111111)
The version for run-lengths instead of runs is
A000005.
The version for normal multisets is 2^(n-1) -
A283353(n-3).
A005811 counts runs in binary expansion.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A098859 counts partitions with distinct multiplicities, ordered
A242882.
A297770 counts distinct runs in binary expansion.
A003242 counts anti-run compositions.
Counting words with all distinct runs:
-
A351202 = permutations of prime factors.
-
Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],!UnsameQ@@Split[#]&]=={}&]],{n,0,15}]
-
\\ here Q(n) is A000009.
Q(n)={polcoef(prod(k=1, n, 1 + x^k + O(x*x^n)), n)}
a(n)={Q(n) + if(n, numdiv(n) - 1) + sum(k=1, (n-1)\3, sum(j=3, (n-1)\k, j%2==1 && n-k*j<>k))} \\ Andrew Howroyd, Feb 15 2022
A351203
Number of integer partitions of n of whose permutations do not all have distinct runs.
Original entry on oeis.org
0, 0, 0, 0, 1, 2, 3, 6, 11, 16, 24, 36, 52, 73, 101, 135, 184, 244, 321, 418, 543, 694, 889, 1127, 1427, 1789, 2242, 2787, 3463, 4276, 5271, 6465, 7921, 9655, 11756, 14254, 17262, 20830, 25102, 30152, 36172, 43270, 51691, 61594, 73300, 87023, 103189, 122099, 144296, 170193, 200497
Offset: 0
The a(4) = 1 through a(9) = 16 partitions:
(211) (221) (411) (322) (332) (441)
(311) (2211) (331) (422) (522)
(21111) (511) (611) (711)
(3211) (3221) (3321)
(22111) (3311) (4221)
(31111) (4211) (4311)
(22211) (5211)
(32111) (22221)
(41111) (32211)
(221111) (33111)
(2111111) (42111)
(51111)
(222111)
(321111)
(2211111)
(3111111)
For example, the partition x = (2,1,1,1,1) has the permutation (1,1,2,1,1), with runs (1,1), (2), (1,1), which are not all distinct, so x is counted under a(6).
The version for run-lengths instead of runs is
A144300.
The version for normal multisets is
A283353.
The Heinz numbers of these partitions are
A351201.
The complement is counted by
A351204.
A005811 counts runs in binary expansion.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A098859 counts partitions with distinct multiplicities, ordered
A242882.
A297770 counts distinct runs in binary expansion.
Counting words with all distinct runs:
-
A351202 = permutations of prime factors.
Cf.
A000041,
A035363,
A047993,
A116608,
A238130 or
A238279,
A325545,
A329746,
A350842,
A351003,
A351004,
A351291.
-
Table[Length[Select[IntegerPartitions[n],MemberQ[Permutations[#],_?(!UnsameQ@@Split[#]&)]&]],{n,0,15}]
-
from sympy.utilities.iterables import partitions
from itertools import permutations, groupby
from collections import Counter
def A351203(n):
c = 0
for s, p in partitions(n,size=True):
for q in permutations(Counter(p).elements(),s):
if max(Counter(tuple(g) for k, g in groupby(q)).values(),default=0) > 1:
c += 1
break
return c # Chai Wah Wu, Oct 16 2023
A382915
Number of integer partitions of n having no permutation with all equal run-lengths.
Original entry on oeis.org
0, 0, 0, 0, 0, 1, 2, 4, 4, 9, 11, 18, 21, 34, 41, 55, 69, 98, 120, 160, 189, 249, 309, 396, 472, 605, 734, 913, 1099, 1371, 1632, 2021, 2406, 2937, 3514, 4251, 5039, 6101, 7221, 8646, 10205, 12209, 14347, 17086, 20041, 23713, 27807, 32803, 38262, 45043, 52477, 61471, 71496
Offset: 0
The partition y = (2,2,1,1,1) has permutations and run-lengths:
(2,2,1,1,1) (2,3)
(2,1,2,1,1) (1,1,1,2)
(2,1,1,2,1) (1,2,1,1)
(2,1,1,1,2) (1,3,1)
(1,2,2,1,1) (1,2,2)
(1,2,1,2,1) (1,1,1,1,1)
(1,2,1,1,2) (1,1,2,1)
(1,1,2,2,1) (2,2,1)
(1,1,2,1,2) (2,1,1,1)
(1,1,1,2,2) (3,2)
Since (1,2,1,2,1) has all equal run-lengths (1,1,1,1,1), y is not counted under a(7).
The a(5) = 1 through a(10) = 11 partitions:
(2111) (3111) (2221) (5111) (3222) (3331)
(21111) (4111) (41111) (6111) (4222)
(31111) (311111) (22221) (7111)
(211111) (2111111) (51111) (61111)
(321111) (421111)
(411111) (511111)
(2211111) (3211111)
(3111111) (4111111)
(21111111) (22111111)
(31111111)
(211111111)
The complement for distinct run-lengths is
A239455, ranked by
A351294.
For distinct instead of equal run-lengths we have
A351293, ranked by
A351295.
The complement is counted by
A383013.
A382857 counts permutations of prime indices with equal run-lengths.
-
Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],SameQ@@Length/@Split[#]&]=={}&]],{n,0,15}]
A384390
Heinz numbers of integer partitions with a unique proper way to choose disjoint strict partitions of each part.
Original entry on oeis.org
5, 7, 21, 22, 26, 33, 35, 39, 102, 114, 130, 154, 165, 170, 190, 195, 231, 238, 255, 285
Offset: 1
The strict partition (7,2,1) with Heinz number 102 can only be properly refined as ((4,3),(2),(1)), so 102 is in the sequence. The other refinement ((7),(2),(1)) is not proper.
The terms together with their prime indices begin:
5: {3}
7: {4}
21: {2,4}
22: {1,5}
26: {1,6}
33: {2,5}
35: {3,4}
39: {2,6}
102: {1,2,7}
114: {1,2,8}
130: {1,3,6}
154: {1,4,5}
165: {2,3,5}
170: {1,3,7}
190: {1,3,8}
195: {2,3,6}
231: {2,4,5}
238: {1,4,7}
255: {2,3,7}
285: {2,3,8}
This is the case of a unique proper choice in
A384322.
These are positions of 1 in
A384389.
A357982 counts strict partitions of each prime index, non-strict
A299200.
-
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y],y],UnsameQ@@#&];
Select[Range[100],Length[pofprop[prix[#]]]==1&]
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