A335452 -id:A335452 - cp's OEIS frontend

A335463 Numbers k such that there exists a permutation of the prime indices of k matching both (1,2,1) and (2,1,2).

36, 72, 90, 100, 108, 126, 144, 180, 196, 198, 200, 216, 225, 234, 252, 270, 288, 300, 306, 324, 342, 350, 360, 378, 392, 396, 400, 414, 432, 441, 450, 468, 484, 500, 504, 522, 525, 540, 550, 558, 576, 588, 594, 600, 612, 630, 648, 650, 666, 675, 676, 684, 700

Offset: 1

Gus Wiseman, Jun 20 2020

nonn

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

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The sequence of terms together with their prime indices begins:
   36: {1,1,2,2}
   72: {1,1,1,2,2}
   90: {1,2,2,3}
  100: {1,1,3,3}
  108: {1,1,2,2,2}
  126: {1,2,2,4}
  144: {1,1,1,1,2,2}
  180: {1,1,2,2,3}
  196: {1,1,4,4}
  198: {1,2,2,5}
  200: {1,1,1,3,3}
  216: {1,1,1,2,2,2}
  225: {2,2,3,3}
  234: {1,2,2,6}
  252: {1,1,2,2,4}
  270: {1,2,2,2,3}
  288: {1,1,1,1,1,2,2}
  300: {1,1,2,3,3}

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Replacing "and" with "or" gives A126706.
Positions of nonzero terms in A335462.
Permutations of prime indices are counted by A008480.
Unsorted prime signature is A124010. Sorted prime signature is A118914.
STC-numbers of permutations of prime indices are A333221.
Patterns matched by standard compositions are counted by A335454.
Cf. A056239, A056986, A112798, A158005, A181796, A333175, A335451, A335452, A335460, A335465.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Select[Permutations[primeMS[#]],MatchQ[#,{_,x_,_,y_,_,x_,_}/;x_,x_,_,y_,_,x_,_}/;x>y]&]!={}&]

A336102 Number of inseparable multisets of size n covering an initial interval of positive integers.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 8, 8, 20, 20, 48, 48, 112, 112, 256, 256, 576, 576, 1280, 1280, 2816, 2816, 6144, 6144, 13312, 13312, 28672, 28672, 61440, 61440, 131072, 131072, 278528, 278528, 589824, 589824, 1245184, 1245184, 2621440, 2621440, 5505024, 5505024, 11534336

Offset: 0

Gus Wiseman, Jul 08 2020

nonn

A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.
Alternatively, a multiset is separable if its greatest multiplicity is greater than the sum of its remaining multiplicities plus one.
Also the number of compositions of n whose greatest part is greater than the sum of its remaining parts plus one. For example, the a(2) = 1 through a(7) = 8 compositions are:
  (2)  (3)  (4)    (5)    (6)      (7)
            (1,3)  (1,4)  (1,5)    (1,6)
            (3,1)  (4,1)  (2,4)    (2,5)
                          (4,2)    (5,2)
                          (5,1)    (6,1)
                          (1,1,4)  (1,1,5)
                          (1,4,1)  (1,5,1)
                          (4,1,1)  (5,1,1)

			The a(2) = 1 through a(7) = 8 multisets:
  {11}  {111}  {1111}  {11111}  {111111}  {1111111}
               {1112}  {11112}  {111112}  {1111112}
               {1222}  {12222}  {111122}  {1111122}
                                {111123}  {1111123}
                                {112222}  {1122222}
                                {122222}  {1222222}
                                {122223}  {1222223}
                                {123333}  {1233333}

Michael De Vlieger, Table of n, a(n) for n = 0..6625
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-4).

The strong (weakly decreasing multiplicities) case is A025065.
The bisection is A049610.
The separable version is A336103.
Sequences covering an initial interval are A000670.
Anti-run compositions are A003242.
Anti-run patterns are A005649.
Separable partitions are A325534.
Inseparable partitions are A325535.
Inseparable factorizations are A333487.
Anti-run compositions are ranked by A333489.
Heinz numbers of inseparable partitions are A335448.
Cf. A001792, A019472, A052841, A106351, A124767, A269134, A292884, A335433, A335126, A335452, A335548.

Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],With[{mx=Max@@#},mx>1+Total[DeleteCases[#,mx,{1},1]]]&]],{n,0,15}]
(* Second program: *)
CoefficientList[Series[x^2*(1 - x) (x + 1)^2/(2 x^2 - 1)^2, {x, 0, 43}], x] (* Michael De Vlieger, Apr 07 2021 *)

a(2*n) = a(2*n + 1) = A049610(n + 1).
a(n) = 2^(n-1) - A336103(n).
A001792 repeated for n > 1. David A. Corneth, Jul 09 2020
From Chai Wah Wu, Apr 07 2021: (Start)
a(n) = 4*a(n-2) - 4*a(n-4) for n > 5.
G.f.: x^2*(1 - x)*(x + 1)^2/(2*x^2 - 1)^2. (End)

A336106 Number of integer partitions of n whose greatest part is at most one more than the sum of the other parts.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 7, 11, 15, 23, 30, 44, 58, 82, 105, 146, 186, 252, 318, 423, 530, 695, 863, 1116, 1380, 1763, 2164, 2738, 3345, 4192, 5096, 6334, 7665, 9459, 11395, 13968, 16765, 20425, 24418, 29588, 35251, 42496, 50460, 60547, 71669, 85628

Offset: 0

Gus Wiseman, Jul 09 2020

nonn

Also the number of separable strong multisets of length n covering an initial interval of positive integers. A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.

			The a(1) = 1 through a(8) = 15 partitions:
  (1)  (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (211)   (221)    (222)     (322)      (332)
                    (1111)  (311)    (321)     (331)      (422)
                            (2111)   (2211)    (421)      (431)
                            (11111)  (3111)    (2221)     (2222)
                                     (21111)   (3211)     (3221)
                                     (111111)  (4111)     (3311)
                                               (22111)    (4211)
                                               (31111)    (22211)
                                               (211111)   (32111)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

The inseparable version is A025065.
The Heinz numbers of these partitions are A335127.
The non-strong version is A336103.
Sequences covering an initial interval are A000670.
Anti-run compositions are A003242.
Anti-run patterns are A005649.
Separable partitions are A325534.
Inseparable partitions are A325535.
Separable factorizations are A335434.
Heinz numbers of separable partitions are A335433.
Cf. A049610, A106351, A292884, A335126, A335433, A335452, A335548, A336102.

Table[Length[Select[IntegerPartitions[n],2*Max@@#<=1+n&]],{n,0,15}]

A333487 Number of inseparable factorizations of n into factors > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jul 01 2020

nonn

A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.

			The a(n) factorizations for n = 4, 16, 96, 144, 64, 192:
  2*2  4*4      2*2*2*12     12*12        8*8          3*4*4*4
       2*2*2*2  2*2*2*2*6    2*2*2*18     4*4*4        2*2*2*24
                2*2*2*2*2*3  2*2*2*2*9    2*2*2*8      2*2*2*2*12
                             2*2*2*2*3*3  2*2*2*2*4    2*2*2*2*2*6
                                          2*2*2*2*2*2  2*2*2*2*3*4
                                                       2*2*2*2*2*2*3

The version for partitions is A325535.
The version for multisets with prescribed multiplicities is A335126.
The separable version is A335434.
Anti-run compositions are A003242.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Patterns contiguously matched by compositions are A335457.
Cf. A106351, A292884, A295370, A333628, A333755, A335463, A335125, A335127, A335407, A335474, A335516, A335838.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Select[Permutations[#],!MatchQ[#,{_,x_,x_,_}]&]=={}&]],{n,100}]

a(n) + A335434(n) = A001055(n).

A347050 Number of factorizations of n that are a twin (x*x) or have an alternating permutation.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 5, 1, 5, 2, 2, 2, 9, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 10, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 15, 1, 2, 4, 4, 2, 5, 1, 10, 4, 2, 1, 11, 2

Offset: 1

Gus Wiseman, Oct 15 2021

nonn

First differs from A348383 at a(216) = 27, A348383(216) = 28.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
These permutations are ordered factorizations of n with no adjacent triples (..., x, y, z, ...) where x <= y <= z or x >= y >= z.
The version without twins for n > 0 is a(n) + 1 if n is a perfect square; otherwise a(n).

			The factorizations for n = 4, 12, 24, 30, 36, 48, 60, 64, 72:
  4    12     24     30     36       48       60       64       72
  2*2  2*6    3*8    5*6    4*9      6*8      2*30     8*8      8*9
       3*4    4*6    2*15   6*6      2*24     3*20     2*32     2*36
       2*2*3  2*12   3*10   2*18     3*16     4*15     4*16     3*24
              2*2*6  2*3*5  3*12     4*12     5*12     2*4*8    4*18
              2*3*4         2*2*9    2*3*8    6*10     2*2*16   6*12
                            2*3*6    2*4*6    2*5*6    2*2*4*4  2*4*9
                            3*3*4    3*4*4    3*4*5             2*6*6
                            2*2*3*3  2*2*12   2*2*15            3*3*8
                                     2*2*3*4  2*3*10            3*4*6
                                              2*2*3*5           2*2*18
                                                                2*3*12
                                                                2*2*3*6
                                                                2*3*3*4
                                                                2*2*2*3*3
The a(270) = 19 factorizations:
  (2*3*5*9)   (5*6*9)   (3*90)   (270)
  (3*3*5*6)   (2*3*45)  (5*54)
  (2*3*3*15)  (2*5*27)  (6*45)
              (2*9*15)  (9*30)
              (3*3*30)  (10*27)
              (3*5*18)  (15*18)
              (3*6*15)  (2*135)
              (3*9*10)
Note that (2*3*3*3*5) is separable but has no alternating permutations.

Partitions not of this type are counted by A344654, ranked by A344653.
Partitions of this type are counted by A344740, ranked by A344742.
The complement is counted by A347706, without twins A348380.
The case without twins is A348379.
Dominates A348383, the separable case.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations.
A008480 counts permutations of prime indices, strict A335489.
A025047 counts alternating or wiggly compositions, ranked by A345167.
A056239 adds up prime indices, row sums of A112798.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
Cf. A038548, A049774, A102726, A119620, A128761, A344614, A347437, A347438, A347458, A348381.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Function[f,Select[Permutations[f],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]!={}]]],{n,100}]

For n > 1, a(n) = A335434(n) + A010052(n).

A349056 Number of weakly alternating permutations of the multiset of prime factors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 4, 1, 1, 2, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 6, 1, 2, 3, 1, 2, 4, 1, 3, 2, 4, 1, 6, 1, 2, 3, 3, 2, 4, 1, 5, 1, 2, 1, 6, 2, 2, 2

Offset: 1

Gus Wiseman, Dec 02 2021

nonn

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. Then a sequence is alternating in the sense of A025047 iff it is a weakly alternating anti-run.

A prime index of n is a number m such that prime(m) divides n. For n > 1, the multiset of prime factors of n is row n of A027746. The prime indices A112798 can also be used.

			The following are the weakly alternating permutations for selected n:
n = 2   6    12    24     48      60     90     120     180
   ----------------------------------------------------------
    2   23   223   2223   22223   2253   2335   22253   22335
        32   232   2232   22232   2325   2533   22325   22533
             322   2322   22322   2523   3253   22523   23253
                   3222   23222   3252   3325   23252   23352
                          32222   3522   3352   25232   25233
                                  5232   3523   32225   25332
                                         5233   32522   32325
                                         5332   35222   32523
                                                52223   33252
                                                52322   33522
                                                        35232
                                                        52323
                                                        53322

Counting all permutations of prime factors gives A008480.
The variation counting anti-run permutations is A335452.
The strong case is A345164, with twins A344606.
Compositions of this type are counted by A349052, also A129852 and A129853.
Compositions not of this type are counted by A349053, ranked by A349057.
The version for patterns is A349058, strong A345194.
The version for ordered factorizations is A349059, strong A348610.
Partitions of this type are counted by A349060, complement A349061.
The complement is counted by A349797.
The non-alternating case is A349798.
A001250 counts alternating permutations, complement A348615.
A003242 counts Carlitz (anti-run) compositions.
A025047 counts alternating or wiggly compositions, ranked by A345167.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A071321 gives the alternating sum of prime factors, reverse A071322.
A344616 gives the alternating sum of prime indices, reverse A316524.
A345165 counts partitions w/o an alternating permutation, ranked by A345171.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A348379 counts factorizations with an alternating permutation.
A349800 counts weakly but not strongly alternating compositions.
Cf. A028234, A051119, A096441, A335433, A335448, A344614, A344652, A344653, A345173, A345192.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Permutations[primeMS[n]],whkQ[#]||whkQ[-#]&]],{n,100}]

A386583 Triangle read by rows where T(n,k) is the number of length k integer partitions of n having a permutation without any adjacent equal parts (separable).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 2, 0, 0, 0, 1, 2, 2, 1, 0, 0, 0, 1, 3, 4, 1, 1, 0, 0, 0, 1, 3, 5, 3, 2, 0, 0, 0, 0, 1, 4, 6, 4, 3, 1, 0, 0, 0, 0, 1, 4, 8, 6, 5, 1, 1, 0, 0, 0, 0, 1, 5, 10, 8, 8, 3, 2, 0, 0, 0, 0, 0, 1, 5, 11, 12, 11, 5, 3, 1, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Aug 03 2025

A multiset is separable iff it has a permutation that is an anti-run, meaning there are no adjacent equal parts.
Separable partitions (A325534) are different from partitions of separable type (A386585).
Are the rows all unimodal?
Some rows are not unimodal: T(200, k=26..30) = 149371873744, 153304102463, 152360653274, 152412869411, 147228477998. - Alois P. Heinz, Aug 04 2025

			Row n = 9 counts the following partitions:
  (9)  (5,4)  (4,3,2)  (3,3,2,1)  (3,2,2,1,1)  (2,2,2,1,1,1)
       (6,3)  (4,4,1)  (4,2,2,1)  (3,3,1,1,1)
       (7,2)  (5,2,2)  (4,3,1,1)  (4,2,1,1,1)
       (8,1)  (5,3,1)  (5,2,1,1)
              (6,2,1)
              (7,1,1)
Triangle begins:
  1
  0  1
  0  1  0
  0  1  1  0
  0  1  1  1  0
  0  1  2  2  0  0
  0  1  2  2  1  0  0
  0  1  3  4  1  1  0  0
  0  1  3  5  3  2  0  0  0
  0  1  4  6  4  3  1  0  0  0
  0  1  4  8  6  5  1  1  0  0  0
  0  1  5 10  8  8  3  2  0  0  0  0
  0  1  5 11 12 11  5  3  1  0  0  0  0
  0  1  6 14 14 15  8  6  1  1  0  0  0  0
  0  1  6 16 19 20 11  9  3  2  0  0  0  0  0
  0  1  7 18 23 27 17 14  5  3  1  0  0  0  0  0
  0  1  7 21 29 34 23 20  9  6  1  1  0  0  0  0  0
  0  1  8 24 34 43 32 28 13 10  3  2  0  0  0  0  0  0
  0  1  8 26 42 53 42 38 20 15  5  3  1  0  0  0  0  0  0
  0  1  9 30 48 66 55 52 28 23  9  6  1  1  0  0  0  0  0  0
  0  1  9 33 58 80 70 68 41 33 14 10  3  2  0  0  0  0  0  0  0
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Separable case of A008284.
Row sums are A325534, ranked by A335433.
For inseparable instead separable we have A386584, sums A325535, ranks A335448.
For separable type instead of separable we have A386585, sums A336106, ranks A335127.
For inseparable type instead of separable we have A386586, sums A025065, ranks A335126.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A124762 gives inseparability of standard compositions, separability A333382.
A239455 counts Look-and-Say partitions, ranks A351294.
A336103 counts normal separable multisets, inseparable A336102.
A351293 counts non-Look-and-Say partitions, ranks A351295.
A386633 counts separable set partitions, row sums of A386635.
A386634 counts inseparable set partitions, row sums of A386636.
Cf. A106351, A111133, A238130, A335434, A386575, A386576, A386580, A386581, A386577.

sepQ[y_]:=Select[Permutations[y],Length[Split[#]]==Length[y]&]!={};
Table[Length[Select[IntegerPartitions[n,{k}],sepQ]],{n,0,15},{k,0,n}]

A335489 Number of strict permutations of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 19 2020

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.

Also the number of (1,1)-avoiding permutations of the prime indices of n.

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Positions of first appearances are A002110 with 2 replaced by 4.
Permutations of prime indices are counted by A008480.
The contiguous version is A335451.
Anti-run permutations of prime indices are counted by A335452.
(1,1,1)-avoiding permutations of prime indices are counted by A335511.
Cf. A056239, A056986, A106356, A112798, A238279, A281188, A333221, A335456, A335460, A335462, A335465.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],!MatchQ[#,{_,x_,_,x_,_}]&]],{n,100}]

If n is squarefree, a(n) = A001221(n)!; otherwise a(n) = 0.

a(n != 4) = A281188(n); a(4) = 0.

A344652 Number of permutations of the prime indices of n with no adjacent triples (..., x, y, z, ...) such that x <= y <= z.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 2, 1, 2, 2, 0, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 0, 2, 1, 5, 1, 0, 2, 2, 2, 3, 1, 2, 2, 1, 1, 5, 1, 2, 2, 2, 1, 0, 1, 2, 2, 2, 1, 1, 2, 1, 2, 2, 1, 7, 1, 2, 2, 0, 2, 5, 1, 2, 2, 5, 1, 2, 1, 2, 2, 2, 2, 5, 1, 0, 0, 2, 1, 7, 2, 2, 2

Offset: 1

Gus Wiseman, Jun 17 2021

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.

			The permutations for n = 2, 6, 8, 30, 36, 60, 180, 210, 360:
  (1)  (12)  (132)  (1212)  (1213)  (12132)  (1324)  (121213)
       (21)  (213)  (2121)  (1312)  (13212)  (1423)  (121312)
             (231)  (2211)  (1321)  (13221)  (1432)  (121321)
             (312)          (2131)  (21213)  (2143)  (131212)
             (321)          (2311)  (21312)  (2314)  (132121)
                            (3121)  (21321)  (2413)  (132211)
                            (3211)  (22131)  (2431)  (212131)
                                    (23121)  (3142)  (213121)
                                    (23211)  (3214)  (213211)
                                    (31212)  (3241)  (221311)
                                    (32121)  (3412)  (231211)
                                    (32211)  (3421)  (312121)
                                             (4132)  (321211)
                                             (4213)
                                             (4231)
                                             (4312)
                                             (4321)

All permutations of prime indices are counted by A008480.
The case of permutations is A049774.
Avoiding (3,2,1) also gives A344606.
The wiggly case is A345164.
A001250 counts wiggly permutations.
A025047 counts wiggly compositions (ascend: A025048, descend: A025049).
A056239 adds up prime indices, row sums of A112798.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A335452 counts anti-run permutations of prime indices.
A345170 counts partitions with a wiggly permutation, ranked by A345172.
A345192 counts non-wiggly compositions, ranked by A345168.
Counting compositions by patterns:
- A102726 avoiding (1,2,3).
- A128761 avoiding (1,2,3) adjacent.
- A335514 matching (1,2,3).
- A344614 avoiding (1,2,3) and (3,2,1) adjacent.
- A344615 weakly avoiding (1,2,3) adjacent.
Cf. A001222, A003242, A056986, A316524, A333213, A335511, A344604, A344653, A344654, A345167, A345173.

Table[Length[Select[Permutations[Flatten[ ConstantArray@@@FactorInteger[n]]],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z]&]],{n,100}]

A386584 Triangle read by rows where T(n,k) is the number of length k>=0 integer partitions of n having no permutation without any adjacent equal parts (inseparable).

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Aug 05 2025

A multiset is inseparable iff it has no anti-run permutations, where an anti-run is a sequence without any adjacent equal parts. Inseparable partitions (A325535) are different from partitions of inseparable type (A386586).

			Row n = 10 counts the following partitions:
  . . 55 . 7111 61111 511111 4111111 31111111 211111111 1111111111
           4222 22222 421111 3211111 22111111
           3331       331111
                      222211
Triangle begins:
  0
  0  0
  0  0  1
  0  0  0  1
  0  0  1  0  1
  0  0  0  0  1  1
  0  0  1  1  1  1  1
  0  0  0  0  2  1  1  1
  0  0  1  0  2  1  2  1  1
  0  0  0  1  2  2  2  2  1  1
  0  0  1  0  3  2  4  2  2  1  1
  0  0  0  0  3  2  4  3  3  2  1  1
  0  0  1  1  3  2  6  4  4  3  2  1  1
  0  0  0  0  4  3  6  5  6  4  3  2  1  1
  0  0  1  0  4  3  9  6  8  5  5  3  2  1  1
  0  0  0  1  4  3  9  7 10  8  6  5  3  2  1  1
  0  0  1  0  5  3 12  8 13  9 10  6  5  3  2  1  1
  0  0  0  0  5  4 12 10 16 12 12  9  7  5  3  2  1  1
  0  0  1  1  5  4 16 11 20 15 17 12 10  7  5  3  2  1  1
  0  0  0  0  6  4 16 13 24 18 21 16 14 10  7  5  3  2  1  1
  0  0  1  0  6  4 20 14 29 21 28 20 19 13 11  7  5  3  2  1  1

Inseparable case of A008284 or A072233.
Row sums are A325535, ranked by A335448.
For separable instead of inseparable we have A386583, sums A325534, ranks A335433.
For separable type we have A386585, sums A336106, ranks A335127.
For inseparable type we have A386586, sums A025065, ranks A335126.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A124762 gives inseparability of standard compositions, separability A333382.
A336103 counts normal separable multisets, inseparable A336102.
A386633 counts separable set partitions, row sums of A386635.
A386634 counts inseparable set partitions, row sums of A386636.
Cf. A106351, A111133, A238130, A239455, A335434, A351293, A386575, A386576, A386580, A386581, A386577.

insepQ[y_]:=Select[Permutations[y],Length[Split[#]]==Length[y]&]=={};
Table[Length[Select[IntegerPartitions[n,{k}],insepQ]],{n,0,15},{k,0,n}]

T(n,k) = A072233(n,k) - A386583(n,k).

cp's OEIS Frontend

A335463 Numbers k such that there exists a permutation of the prime indices of k matching both (1,2,1) and (2,1,2).

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A336102 Number of inseparable multisets of size n covering an initial interval of positive integers.

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A336106 Number of integer partitions of n whose greatest part is at most one more than the sum of the other parts.

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A333487 Number of inseparable factorizations of n into factors > 1.

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A347050 Number of factorizations of n that are a twin (x*x) or have an alternating permutation.

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A349056 Number of weakly alternating permutations of the multiset of prime factors of n.

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A386583 Triangle read by rows where T(n,k) is the number of length k integer partitions of n having a permutation without any adjacent equal parts (separable).

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A335489 Number of strict permutations of the prime indices of n.

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