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}]
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.
A383098
Number of integer partitions of n having at least one permutation with all equal run-sums.
Original entry on oeis.org
1, 1, 2, 2, 4, 2, 7, 2, 7, 5, 7, 2, 19, 2, 7, 8, 14, 2, 27, 2, 24, 8, 7, 2, 58, 5, 7, 13, 30, 2, 72, 2, 38, 8, 7, 8, 135, 2, 7, 8, 91, 2, 112, 2, 45, 38, 7, 2, 258, 5, 51, 8, 54, 2, 208, 8, 143, 8, 7, 2, 525, 2, 7, 44, 153, 8, 256, 2, 75, 8, 136, 2, 891, 2, 7, 57, 87, 8
Offset: 0
The partition (4,4,4,2,2,1,1,1,1) has permutations (4,2,2,4,1,1,1,1,4) and (4,1,1,1,1,4,2,2,4) so is counted under a(20).
The a(1) = 1 through a(10) = 7 partitions (A=10):
1 2 3 4 5 6 7 8 9 A
11 111 22 11111 33 1111111 44 333 55
211 222 422 33111 22222
1111 2211 2222 3111111 511111
3111 41111 111111111 2221111
21111 221111 22111111
111111 11111111 1111111111
For distinct instead of equal run-sums we appear to have
A382427.
For run-lengths instead of sums we have
A383013, ranked by complement of
A382879.
These partitions are ranked by
A383110.
Counting and ranking partitions by run-lengths and run-sums:
Cf.
A006171,
A329738,
A353832,
A353839,
A353850,
A353932,
A354584,
A382076,
A382857,
A382876,
A383094,
A383112.
-
Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],SameQ@@Total/@Split[#]&]!={}&]],{n,0,15}]
A383110
Numbers whose prime indices have a permutation with all equal run-sums.
Original entry on oeis.org
1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 40, 41, 43, 47, 48, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169, 173
Offset: 1
The prime indices of 144 are {1,1,1,1,2,2}, with permutations with equal run sums (1,1,1,1,2,2), (1,1,2,1,1,2), (2,1,1,2,1,1), (2,2,1,1,1,1), 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}
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}
For distinct run-sums we appear to have complement of
A381636 (counted by
A381717).
These are the positions of positive terms in
A382877.
For run-lengths instead of sums we have complement of
A382879, counted by
A383013.
For more than one choice we have
A383015.
Partitions of this type are counted by
A383098.
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}]
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}]
A384904
Number of integer partitions of n with all equal lengths of maximal runs of consecutive parts decreasing by 1 but not by 0.
Original entry on oeis.org
1, 1, 2, 3, 4, 5, 9, 9, 14, 17, 23, 25, 40, 41, 59, 68, 92, 99, 140, 151, 204, 229, 296, 328, 433, 476, 606, 685, 858, 955, 1203, 1336, 1654, 1858, 2266, 2537, 3102, 3453, 4169, 4680, 5611, 6262, 7495, 8358, 9927, 11105, 13096, 14613, 17227, 19179, 22459
Offset: 0
The partition (6,5,5,4,2,1) has maximal runs ((6,5),(5,4),(2,1)), with lengths (2,2,2), so is counted under a(23).
The a(1) = 1 through a(8) = 14 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (43) (44)
(111) (31) (41) (42) (52) (53)
(1111) (311) (51) (61) (62)
(11111) (222) (331) (71)
(321) (511) (422)
(411) (4111) (611)
(3111) (31111) (2222)
(111111) (1111111) (3221)
(3311)
(5111)
(41111)
(311111)
(11111111)
For subsets instead of strict partitions we have
A243815, distinct lengths
A384175.
For distinct instead of equal lengths we have
A384882, counting gaps of 0
A384884.
-
Table[Length[Select[IntegerPartitions[n],SameQ@@Length/@Split[#,#2==#1-1&]&]],{n,0,30}]
-
A_q(N) = {Vec(1+sum(k=1,floor(-1/2+sqrt(2+2*N)), sum(i=1,(N/(k*(k+1)/2))+1, q^((k*i*(2+i*(k-1)))/2)/(1-q^(k*i))*prod(j=1,i-1, 1 + q^(2*k*j)/(1 - q^(k*j))))) + O('q^(N+1)))} \\ John Tyler Rascoe, Aug 20 2025
A383111
Number of integer partitions of n having more than one permutation with all distinct run-lengths.
Original entry on oeis.org
0, 0, 0, 0, 1, 3, 3, 8, 9, 13, 17, 26, 27, 43, 51, 61, 78, 103, 115, 153, 174, 213, 255, 316, 354, 442, 508, 610, 701, 848, 950, 1153, 1303, 1539, 1750, 2075, 2318, 2738, 3081
Offset: 0
The partition (2,1,1) has two permutations with all distinct run-lengths: (1,1,2), (2,1,1), so it is counted under a(4).
The a(4) = 1 through a(9) = 13 partitions:
(211) (221) (411) (322) (332) (441)
(311) (3111) (331) (422) (522)
(2111) (21111) (511) (611) (711)
(2221) (5111) (3222)
(4111) (22211) (6111)
(22111) (41111) (22221)
(31111) (221111) (33111)
(211111) (311111) (51111)
(2111111) (222111)
(411111)
(2211111)
(3111111)
(21111111)
For equal instead of distinct run-lengths we have
A383090, ranks
A383089.
These partitions are ranked by
A383113 = positions of terms > 1 in
A382771.
Cf.
A047993,
A329739,
A381541,
A381636,
A381717,
A382857,
A382915,
A383013,
A383092,
A383094,
A383097.
-
Table[Length[Select[IntegerPartitions[n], Length[Select[Permutations[#], UnsameQ@@Length/@Split[#]&]]>1&]],{n,0,15}]
A384882
Number of integer partitions of n with all distinct lengths of maximal runs of consecutive parts decreasing by 1 but not by 0.
Original entry on oeis.org
1, 1, 1, 2, 2, 3, 2, 5, 4, 5, 6, 9, 7, 12, 12, 11, 16, 18, 17, 25, 25, 23, 33, 35, 36, 42, 52, 45, 58, 64, 60, 77, 91, 79, 109, 108, 105, 129, 149, 134, 170, 179, 177, 213, 236, 208, 275, 281, 282, 323, 359, 330, 410, 433, 440, 474, 541, 508, 614, 631, 635
Offset: 0
The partition (6,5,5,5,3,2) has maximal runs ((6,5),(5),(5),(3,2)), with lengths (2,1,1,2), so is not counted under a(26).
The partition (6,5,5,5,4,3,2) has maximal runs ((6,5),(5),(5,4,3,2)), with lengths (2,1,4), so is counted under a(30).
The a(1) = 1 through a(13) = 12 partitions:
1 2 3 4 5 6 7 8 9 A B C D
21 211 32 321 43 332 54 433 65 543 76
221 322 431 432 532 443 651 544
421 521 621 541 542 732 643
3211 3321 721 632 921 652
4321 821 6321 832
4322 43221 A21
5321 4432
43211 5431
7321
43321
432211
For subsets instead of strict partitions we have
A384175, equal lengths
A243815.
For equal instead of distinct lengths we have
A384904, strict case
A384886.
-
Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#,#2==#1-1&]&]],{n,0,30}]
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