A381433 -id:A381433 - cp's OEIS frontend

A382878 Set of positions of first appearances in A382857 (permutations of prime indices with equal run-lengths).

1, 6, 24, 30, 36, 180, 210, 360, 420, 720, 1080, 1260, 1800, 2160, 2310, 2520, 3600, 4620, 5040, 5400, 6300, 7560, 10800, 12600, 13860, 15120, 21600, 25200, 25920, 27000, 27720, 30030, 32400, 37800, 44100, 45360, 46656, 50400, 54000, 55440, 60060, 60480, 64800

Offset: 1

Gus Wiseman, Apr 09 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

			The permutations for n = 6, 720, 36, 25920, 30:
  (1,2)  (1,2,1,2,1,3,1)  (1,1,2,2)  (1,2,1,2,1,2,1,2,1,3,1)  (1,2,3)
  (2,1)  (1,2,1,3,1,2,1)  (1,2,1,2)  (1,2,1,2,1,2,1,3,1,2,1)  (1,3,2)
         (1,3,1,2,1,2,1)  (2,1,2,1)  (1,2,1,2,1,3,1,2,1,2,1)  (2,1,3)
                          (2,2,1,1)  (1,2,1,3,1,2,1,2,1,2,1)  (2,3,1)
                                     (1,3,1,2,1,2,1,2,1,2,1)  (3,1,2)
                                                              (3,2,1)
The terms together with their prime indices begin:
      1: {}
      6: {1,2}
     24: {1,1,1,2}
     30: {1,2,3}
     36: {1,1,2,2}
    180: {1,1,2,2,3}
    210: {1,2,3,4}
    360: {1,1,1,2,2,3}
    420: {1,1,2,3,4}
    720: {1,1,1,1,2,2,3}
   1080: {1,1,1,2,2,2,3}
   1260: {1,1,2,2,3,4}
   1800: {1,1,1,2,2,3,3}
   2160: {1,1,1,1,2,2,2,3}
   2310: {1,2,3,4,5}
   2520: {1,1,1,2,2,3,4}
   3600: {1,1,1,1,2,2,3,3}

Positions of first appearances in A382857 (zeros A382879), by signature A382858.
For distinct run-lengths we have A382772, firsts of A382771 (by signature A382773).
A140690 lists numbers whose binary expansion has equal run-lengths, distinct A044813.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A000720, A001221, A001222, A003242, A048767, A098859, A130091, A238130, A305936, A351013, A351202, A382876.

y=Table[Length[Select[Permutations[Join@@ConstantArray@@@FactorInteger[n]],SameQ@@Length/@Split[#]&]],{n,0,1000}];
fip[y_]:=Select[Range[Length[y]],!MemberQ[Take[y,#-1],y[[#]]]&];
fip[Rest[y]]

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

Gus Wiseman, Apr 19 2025

nonn

			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)

For no choices we have A382915, ranks A382879.
For at least one choice we have A383013, for run-sums A383098, ranks A383110.
The complement is A383090, ranks A383089.
Partitions of this type are ranked by A383091 = positions of terms <= 1 in A382857.
For a unique choice we have A383094, ranks A383112.
For run-sums instead of lengths we have A383095 + A383096, ranks A383099 \/ A383100.
The complement for run-sums is A383097, ranks A383015, positions of terms > 1 in A382877.
Cf. A047966, A072774, A098859, A100471, A100881, A100882, A100883, A304442.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A006171, A047993, A362608, A381871, A382076, A382771, A383111.

Table[Length[Select[IntegerPartitions[n],Length[Select[Permutations[#],SameQ@@Length/@Split[#]&]]<=1&]],{n,0,15}]

a(n) = A382915(n) + A383094(n).

More terms from Bert Dobbelaere, Apr 26 2025

A383507 Number of Wilf and conjugate Wilf integer partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 6, 7, 9, 12, 14, 19, 20, 27, 30, 31, 40, 50, 56, 68, 76, 86, 112, 126, 139, 170, 197, 216, 251, 297, 317, 378, 411, 466, 521, 607, 621, 745, 791, 892, 975, 1123, 1163, 1366, 1439, 1635, 1757, 2021, 2080, 2464, 2599, 2882, 3116, 3572, 3713

Offset: 0

Gus Wiseman, May 14 2025

nonn

An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

			The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (1111)  (11111)  (222)     (331)      (332)
                                     (411)     (511)      (611)
                                     (3111)    (4111)     (2222)
                                     (111111)  (31111)    (5111)
                                               (1111111)  (41111)
                                                          (311111)
                                                          (11111111)

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

A048768 gives Look-and-Say fixed points, counted by A217605.
A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
A239455 counts Look-and-Say partitions, complement A351293.
A325349 counts partitions with distinct augmented differences, ranks A325366.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A381431 is the section-sum transform, union A381432, complement A381433.
A383534 gives 0-prepended differences by rank, see A325351.
A383709 counts Wilf partitions with distinct 0-appended differences.
Cf. A047966, A048767, A111133, A320348, A325324, A325325, A325367, A325368, A325388, A351294, A383506.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#] && UnsameQ@@DeleteCases[Differences[Append[#,0]],0]&]],{n,0,30}]

These partitions have Heinz numbers A130091 /\ A383512.

A383514 Heinz numbers of non Wilf section-sum partitions.

Original entry on oeis.org

10, 14, 15, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 129, 130, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 170, 177, 178, 182, 183, 185, 187, 190

Offset: 1

Gus Wiseman, May 18 2025

nonn

First differs from A384007 in having 1000.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
An integer partition is Wilf iff its multiplicities are all different, ranked by A130091.
An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.

			The terms together with their prime indices begin:
    10: {1,3}    57: {2,8}      94: {1,15}
    14: {1,4}    58: {1,10}     95: {3,8}
    15: {2,3}    62: {1,11}    100: {1,1,3,3}
    22: {1,5}    65: {3,6}     106: {1,16}
    26: {1,6}    69: {2,9}     111: {2,12}
    33: {2,5}    74: {1,12}    115: {3,9}
    34: {1,7}    77: {4,5}     118: {1,17}
    35: {3,4}    82: {1,13}    119: {4,7}
    38: {1,8}    85: {3,7}     122: {1,18}
    39: {2,6}    86: {1,14}    123: {2,13}
    46: {1,9}    87: {2,10}    129: {2,14}
    51: {2,7}    91: {4,6}     130: {1,3,6}
    55: {3,5}    93: {2,11}    133: {4,8}

Ranking sequences are shown in parentheses below.
For Look-and-Say instead of section-sum we have A351592 (A384006).
These partitions are counted by A383506.
The Look-and-Say case is A383511 (A383518).
For Wilf instead of non Wilf we have A383519 (A383520).
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A336866 counts non Wilf partitions (A130092), conjugate (A383513).
A381431 is the section-sum transform.
A383508 counts partitions that are both Look-and-Say and section-sum (A383515).
A383509 counts partitions that are Look-and-Say but not section-sum (A383516).
A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
Cf. A000720, A001223, A048767, A051903, A212166, A320348, A325366, A325368, A383531.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],disjointFamilies[conj[prix[#]]]!={}&&!UnsameQ@@Last/@FactorInteger[#]&]

A383519 Number of section-sum partitions of n that have all distinct multiplicities (Wilf).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 6, 7, 9, 12, 14, 19, 21, 27, 30, 33, 41, 50, 57, 68, 79, 89, 112, 126, 144, 172, 198, 220, 257, 298, 327, 383, 423, 477, 533, 621, 650, 760, 816, 920, 1013

Offset: 0

Gus Wiseman, May 19 2025

An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.

An integer partition is Wilf iff its multiplicities are all different (ranked by A130091).

			The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (1111)  (11111)  (222)     (331)      (332)
                                     (411)     (511)      (611)
                                     (3111)    (4111)     (2222)
                                     (111111)  (31111)    (5111)
                                               (1111111)  (41111)
                                                          (311111)
                                                          (11111111)

Ranking sequences are shown in parentheses below.
For Look-and-Say instead of section-sum we have A098859 (A130091), conjugate (A383512).
For non Wilf instead of Wilf we have A383506 (A383514).
These partitions are ranked by (A383520).
A000041 counts integer partitions, strict A000009.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A336866 counts non Wilf partitions (A130092), conjugate (A383513).
Cf. A047966, A320348, A325324, A325325, A351592, A353837, A381431, A383530, A383709, A384006.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],disjointFamilies[conj[#]]!={}&&UnsameQ@@Length/@Split[#]&]],{n,0,15}]

A386587 Number of ways to choose a pairwise disjoint family of strict integer partitions, one of each exponent in the prime factorization of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 0, 2, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 3, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 4, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 0, 1, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Aug 06 2025

nonn

First differs from A382525 at a(216) = 1, A382525(216) = 2.

			The prime exponents of 864 = 2^5 * 3^3 are (5,3), with disjoint families {{3},{5}}, {{3},{1,4}}, {{5},{1,2}}, so a(864) = 3.

Positions of positive terms are A351294, conjugate A381432.
Positions of 0 are A351295, conjugate A381433.
For ordered set partitions we have A382525.
Positions of first appearances are A382775.
The separable case is A386575.
The inseparable case is A386582, see A386632.
A000110 counts set partitions, ordered A000670.
A003242 and A335452 count separations, ranks A333489.
A239455 counts Look-and-Say partitions, complement A351293.
A279790 counts disjoint families on strongly normal multisets.
A325534 counts separable multisets, ranks A335433, sums of A386583.
A325535 counts inseparable multisets, ranks A335448, sums of A386584.
A386633 counts separable set partitions, row sums of A386635.
A386634 counts inseparable set partitions, row sums of A386636.
Cf. A001221, A001222, A008480, A025065, A051903, A051904, A056239, A130091, A336106, A373957, A386580.

disjointFamilies[y_]:=Union[Sort/@Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&]];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[disjointFamilies[prix[n]]],{n,100}]

A382772 Set of positions of first appearances in A382771 (permutations of prime indices with distinct run-lengths).

Original entry on oeis.org

1, 6, 12, 96, 360, 1536, 3456, 5184, 5760, 6144, 7776, 13824, 23040, 24576, 55296, 62208, 92160

Offset: 1

Gus Wiseman, Apr 09 2025

			The permutations for n = 12, 96, 360, 1536:
  (1,1,2)  (1,1,1,1,1,2)  (1,1,1,2,2,3)  (1,1,1,1,1,1,1,1,1,2)
  (2,1,1)  (1,1,1,2,1,1)  (1,1,1,3,2,2)  (1,1,1,1,1,1,1,2,1,1)
           (1,1,2,1,1,1)  (2,2,1,1,1,3)  (1,1,1,1,1,1,2,1,1,1)
           (2,1,1,1,1,1)  (2,2,3,1,1,1)  (1,1,1,1,1,2,1,1,1,1)
                          (3,1,1,1,2,2)  (1,1,1,1,2,1,1,1,1,1)
                          (3,2,2,1,1,1)  (1,1,1,2,1,1,1,1,1,1)
                                         (1,1,2,1,1,1,1,1,1,1)
                                         (2,1,1,1,1,1,1,1,1,1)

Positions of first appearances in A382771, by signature A382773.
For equal run-lengths we have A382878, firsts of A382857, zeros A382879.
A044813 lists numbers whose binary expansion has distinct run-lengths, equal A140690.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A328592 lists numbers whose binary form has distinct runs of ones, equal A164707.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A000720, A001221, A001222, A003242, A048767, A130091, A238130, A305936, A351013, A351202, A382876.

y=Table[Length[Select[Permutations[Join@@ConstantArray@@@FactorInteger[n]],UnsameQ@@Length/@Split[#]&]],{n,0,100000}];
fip[y_]:=Select[Range[Length[y]],!MemberQ[Take[y,#-1],y[[#]]]&];
fip[Rest[y]]

A384393 Heinz numbers of integer partitions with more than one proper way to choose disjoint strict partitions of each part.

Original entry on oeis.org

11, 13, 17, 19, 23, 25, 29, 31, 34, 37, 38, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 121, 122, 123, 127, 129, 131, 133, 134

Offset: 1

Gus Wiseman, Jun 02 2025

nonn

By "proper" we exclude the case of all singletons, which is disjoint when n is squarefree.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The prime indices of 275 are {3,3,5}, with a total of 2 proper choices: ((3),(2,1),(5)) and ((2,1),(3),(5)), so 275 is in the sequence.
The terms together with their prime indices begin:
    11: {5}      51: {2,7}      82: {1,13}
    13: {6}      53: {16}       83: {23}
    17: {7}      55: {3,5}      85: {3,7}
    19: {8}      57: {2,8}      86: {1,14}
    23: {9}      58: {1,10}     87: {2,10}
    25: {3,3}    59: {17}       89: {24}
    29: {10}     61: {18}       91: {4,6}
    31: {11}     62: {1,11}     93: {2,11}
    34: {1,7}    65: {3,6}      94: {1,15}
    37: {12}     67: {19}       95: {3,8}
    38: {1,8}    69: {2,9}      97: {25}
    41: {13}     71: {20}      101: {26}
    43: {14}     73: {21}      103: {27}
    46: {1,9}    74: {1,12}    106: {1,16}
    47: {15}     77: {4,5}     107: {28}
    49: {4,4}    79: {22}      109: {29}

Without "proper" we get A384321 (strict A384322), counted by A384317 (strict A384318).
The case of no choices is A384349, counted by A384348.
These are positions of terms > 1 in A384389.
The case of a unique proper choice is A384390, counted by A384319.
Partitions of this type are counted by A384395.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say partitions, ranks A351294 or A381432.
A279790 and A279375 count ways to choose disjoint strict partitions of prime indices.
A351293 counts non-Look-and-Say partitions, ranks A351295 or A381433.
Cf. A382912, A383710, A382913, A383708, A383533.
Cf. A179009, A357982, A381454, A382525, A383706, A383707, A384320, A384323.

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&]

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

Gus Wiseman, Apr 20 2025

nonn

First differs from A177425, A182854, A367589 in having 216.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			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 exactly one permutation we have A000961, counted by A000005.
For no choices we have A351293, counted by A351295, conjugate A381433, equal A382879.
For at least one choice we have A351294, conjugate A381432, counted by A239455.
These are positions of terms > 1 in A382771, firsts A382772, equal A382878.
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.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct run-lengths (ordered A242882), ranks A130091.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
Cf. A000720, A001221, A001222, A047966, A048767, A351013, A351202, A381435, A382876, A383090, A383091, A383092, A383112.

Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], UnsameQ@@Length/@Split[#]&]]>1&]

The complement is A000961 \/ A351293, counted by A000005 + A351295.

A382774 Number of ways to permute the prime indices of n! so that the run-lengths are all different.

Original entry on oeis.org

1, 1, 1, 0, 2, 0, 6, 0, 0, 0, 96, 0

Offset: 0

Gus Wiseman, Apr 09 2025

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

			The prime indices of 24 are {1,1,1,2}, with permutations (1,1,1,2) and (2,1,1,1), so a(4) = 2.

For anti-run permutations we have A335407, see also A335125, A382858.
This is the restriction of A382771 to the factorials A000142, equal A382857.
A022559 counts prime indices of n!, sum A081401.
A044813 lists numbers whose binary expansion has distinct run-lengths, equal A140690.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A328592 lists numbers whose binary form has distinct runs of ones, equal A164707.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A000720, A001221, A001222, A003242, A048767, A130091, A181821, A351013, A351202, A360015, A382773, A382879.

Table[Length[Select[Permutations[prix[n!]],UnsameQ@@Length/@Split[#]&]],{n,0,6}]

a(n) = A382771(n!).

cp's OEIS Frontend

A382878 Set of positions of first appearances in A382857 (permutations of prime indices with equal run-lengths).

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A383092 Number of integer partitions of n having at most one permutation with all equal run-lengths.

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A383507 Number of Wilf and conjugate Wilf integer partitions of n.

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A383514 Heinz numbers of non Wilf section-sum partitions.

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A383519 Number of section-sum partitions of n that have all distinct multiplicities (Wilf).

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A386587 Number of ways to choose a pairwise disjoint family of strict integer partitions, one of each exponent in the prime factorization of n.

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A382772 Set of positions of first appearances in A382771 (permutations of prime indices with distinct run-lengths).

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A384393 Heinz numbers of integer partitions with more than one proper way to choose disjoint strict partitions of each part.

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A383113 Numbers whose prime indices have more than one permutation with all distinct run-lengths.

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