A336106 -id:A336106 - cp's OEIS frontend

A239455 Number of Look-and-Say partitions of n; see Comments.

0, 1, 2, 2, 4, 5, 7, 10, 13, 16, 21, 28, 33, 45, 55, 65, 83, 105, 121, 155, 180, 217, 259, 318, 362, 445, 512, 614, 707, 850, 958, 1155, 1309, 1543, 1754, 2079, 2327, 2740, 3085, 3592, 4042, 4699, 5253, 6093, 6815, 7839, 8751, 10069, 11208, 12832, 14266, 16270

Offset: 0

Clark Kimberling and Peter J. C. Moses, Mar 19 2014

nonn

Suppose that p = x(1) >= x(2) >= ... >= x(k) is a partition of n. Let y(1) > y(2) > ... > y(h) be the distinct parts of p, and let m(i) be the multiplicity of y(i) for 1 <= i <= h. Then we can "look" at p as "m(1) y(1)'s and m(2) y(2)'s and ... m(h) y(h)'s". Reversing the m's and y's, we can then "say" the Look-and-Say partition of p, denoted by LS(p). The name "Look-and-Say" follows the example of Look-and-Say integer sequences (e.g., A005150). As p ranges through the partitions of n, LS(p) ranges through all the Look-and-Say partitions of n. The number of these is A239455(n).
The Look-and-Say array is distinct from the Wilf array, described at A098859; for example, the number of Look-and-Say partitions of 9 is A239455(9) = 16, whereas the number of Wilf partitions of 9 is A098859(9) = 15. The Look-and-Say partition of 9 which is not a Wilf partition of 9 is [2,2,2,1,1,1].
Conjecture: a partition is Look-and-Say iff it has a permutation with all distinct run-lengths. For example, the partition y = (2,2,2,1,1,1) has the permutation (2,2,1,1,1,2), with run-lengths (2,3,1), which are all distinct, so y is counted under a(9). - Gus Wiseman, Aug 11 2025
Also the number of integer partitions y of n such that there is a pairwise disjoint way to choose a strict integer partition of each multiplicity (or run-length) of y. - Gus Wiseman, Aug 11 2025

			The 11 partitions of 6 generate 7 Look-and-Say partitions as follows:
6 -> 111111
51 -> 111111
42 -> 111111
411 -> 21111
33 -> 222
321 -> 111111
3111 -> 3111
222 -> 33
2211 -> 222
21111 -> 411
111111 -> 6,
so that a(6) counts these 7 partitions: 111111, 21111, 222, 3111, 33, 411, 6.

These include all Wilf partitions, counted by A098859, ranked by A130091.
These partitions are listed by A239454 in graded reverse-lex order.
Non-Wilf partitions are counted by A336866, ranked by A130092.
A variant for runs is A351204, complement A351203.
The complement is counted by A351293, apparently ranked by A351295, conjugate A381433.
These partitions appear to be ranked by A351294, conjugate A381432.
The non-Wilf case is counted by A351592.
For normal multisets we appear to have A386580, complement A386581.
A000110 counts set partitions, ordered A000670.
A000569 = graphical partitions, complement A339617.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A181819 = Heinz number of the prime signature of n (prime shadow).
A279790 counts disjoint families on strongly normal multisets.
A329738 = compositions with all equal run-lengths.
A386583 counts separable partitions, sums A325534, ranks A335433.
A386584 counts inseparable partitions, sums A325535, ranks A335448.
A386585 counts separable type partitions, sums A336106, ranks A335127.
A386586 counts inseparable type partitions, sums A386638 or A025065, ranks A335126.
Counting words with all distinct run-lengths:
- A032020 = binary expansions, for runs A351018, ranked by A044813.
- A329739 = compositions, for runs A351013, ranked by A351596.
- A351017 = binary words, for runs A351016.
- A351292 = patterns, for runs A351200.
Cf. A000041, A008284, A047966, A182857, A225485, A297770, A304660, A305563, A329746, A351201, A351202, A351291.

LS[part_List] := Reverse[Sort[Flatten[Map[Table[#[[2]], {#[[1]]}] &, Tally[part]]]]]; LS[n_Integer] := #[[Reverse[Ordering[PadRight[#]]]]] &[DeleteDuplicates[Map[LS, IntegerPartitions[n]]]]; TableForm[t = Map[LS[#] &, Range[10]]](*A239454,array*)
Flatten[t](*A239454,sequence*)
Map[Length[LS[#]] &, Range[25]](*A239455*)
(* Peter J. C. Moses, Mar 18 2014 *)
disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
Table[Length[Select[IntegerPartitions[n],Length[disjointFamilies[#]]>0&]],{n,0,10}] (* Gus Wiseman, Aug 11 2025 *)

A336103 Number of separable multisets of size n covering an initial interval of positive integers.

Original entry on oeis.org

1, 1, 1, 3, 5, 13, 24, 56, 108, 236, 464, 976, 1936, 3984, 7936, 16128, 32192, 64960, 129792, 260864, 521472, 1045760, 2091008, 4188160, 8375296, 16763904, 33525760, 67080192, 134156288, 268374016, 536739840, 1073610752, 2147205120, 4294688768, 8589344768, 17179279360, 34358493184

Offset: 0

Gus Wiseman, Jul 09 2020

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. Hence a(n) is the number of compositions of n whose greatest part is at most one more than the sum of its other parts. For example, the a(1) = 1 through a(5) = 13 compositions are:
  (1)  (11)  (12)   (22)    (23)
             (21)   (112)   (32)
             (111)  (121)   (113)
                    (211)   (122)
                    (1111)  (131)
                            (212)
                            (221)
                            (311)
                            (1112)
                            (1121)
                            (1211)
                            (2111)
                            (11111)

			The a(1) = 1 through a(5) = 13 separable multisets:
  {1}  {1,2}  {1,1,2}  {1,1,2,2}  {1,1,1,2,2}
              {1,2,2}  {1,1,2,3}  {1,1,1,2,3}
              {1,2,3}  {1,2,2,3}  {1,1,2,2,2}
                       {1,2,3,3}  {1,1,2,2,3}
                       {1,2,3,4}  {1,1,2,3,3}
                                  {1,1,2,3,4}
                                  {1,2,2,2,3}
                                  {1,2,2,3,3}
                                  {1,2,2,3,4}
                                  {1,2,3,3,3}
                                  {1,2,3,3,4}
                                  {1,2,3,4,4}
                                  {1,2,3,4,5}

Michael De Vlieger, Table of n, a(n) for n = 0..3322
Index entries for linear recurrences with constant coefficients, signature (2,4,-8,-4,8).

The inseparable version is A336102.
The strong (weakly decreasing multiplicities) case is A336106.
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, A025065, A049610, A052841, A106351, A269134, A292884, A335126, A335433, A335452.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
sepQ[m_]:=Select[Permutations[m],!MatchQ[#,{_,x_,x_,_}]&]!={};
Table[Length[Select[allnorm[n],sepQ]],{n,0,5}]
(* or *)
Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],With[{mx=Max@@#},mx<=1+Total[DeleteCases[#,mx,{1},1]]]&]],{n,0,15}] (* or *)
CoefficientList[Series[(x - 1) (2 x^5 + 7 x^4 - 5 x^2 + 1)/((2 x - 1) (2 x^2 - 1)^2), {x, 0, 36}], x] (* Michael De Vlieger, Apr 07 2021 *)

a(n) = 2^(n-1) - (floor(n/2)+1) * 2^(floor(n/2)-2) for n >= 2. - David A. Corneth, Jul 09 2020
From Chai Wah Wu, Apr 07 2021: (Start)
a(n) = 2*a(n-1) + 4*a(n-2) - 8*a(n-3) - 4*a(n-4) + 8*a(n-5) for n > 6.
G.f.: (x - 1)*(2*x^5 + 7*x^4 - 5*x^2 + 1)/((2*x - 1)*(2*x^2 - 1)^2). (End)

a(26)-a(36) from David A. Corneth, Jul 09 2020

A349055 Number of multisets of size n that have an alternating permutation and cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 1, 3, 5, 12, 24, 52, 108, 224, 464, 944, 1936, 3904, 7936, 15936, 32192, 64512, 129792, 259840, 521472, 1043456, 2091008, 4183040, 8375296, 16752640, 33525760, 67055616, 134156288, 268320768, 536739840, 1073496064, 2147205120, 4294443008, 8589344768

Offset: 0

Gus Wiseman, Dec 12 2021

nonn

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). Alternating permutations of multisets are a generalization of alternating or up-down permutations of {1..n}.

The multisets that have an alternating permutation are those which have no part with multiplicity greater than floor(n/2) except for odd n when either the smallest or largest part can have multiplicity ceiling(n/2). - Andrew Howroyd, Jan 13 2024

			The multiset {1,2,2,3} has alternating permutations (2,1,3,2), (2,3,1,2), so is counted under a(4).
The a(1) = 1 through a(5) = 12 multisets:
  {1}  {1,2}  {1,1,2}  {1,1,2,2}  {1,1,1,2,2}
              {1,2,2}  {1,1,2,3}  {1,1,1,2,3}
              {1,2,3}  {1,2,2,3}  {1,1,2,2,2}
                       {1,2,3,3}  {1,1,2,2,3}
                       {1,2,3,4}  {1,1,2,3,3}
                                  {1,1,2,3,4}
                                  {1,2,2,3,3}
                                  {1,2,2,3,4}
                                  {1,2,3,3,3}
                                  {1,2,3,3,4}
                                  {1,2,3,4,4}
                                  {1,2,3,4,5}
As compositions:
  (1)  (1,1)  (1,2)    (2,2)      (2,3)
              (2,1)    (1,1,2)    (3,2)
              (1,1,1)  (1,2,1)    (1,1,3)
                       (2,1,1)    (1,2,2)
                       (1,1,1,1)  (2,1,2)
                                  (2,2,1)
                                  (3,1,1)
                                  (1,1,1,2)
                                  (1,1,2,1)
                                  (1,2,1,1)
                                  (2,1,1,1)
                                  (1,1,1,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Wikipedia, Alternating permutation

The strong inseparable case is A025065.
A separable instead of alternating version is A336103, complement A336102.
The case of weakly decreasing multiplicities is A336106.
The version for non-twin partitions is A344654, ranked by A344653.
The complement for non-twin partitions is A344740, ranked by A344742.
The complement for partitions is A345165, ranked by A345171.
The version for partitions is A345170, ranked by A345172.
The version for factorizations is A348379, complement A348380.
The complement (still covering an initial interval) is counted by A349050.
A000670 counts sequences covering an initial interval, anti-run A005649.
A001250 counts alternating permutations, complement A348615.
A003242 counts Carlitz (anti-run) compositions, ranked by A333489.
A025047 = alternating compositions, ranked by A345167, also A025048/A025049.
A049774 counts permutations avoiding the consecutive pattern (1,2,3).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
Cf. A019472, A052841, A096441, A106351, A106356, A335125, A335127, A344614, A347050, A347706, A348610, A348613, A349054.

allnorm[n_]:=If[n<=0,{{}},Function[s, Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
wigQ[y_]:=Or[Length[y]==0, Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[allnorm[n], Select[Permutations[#],wigQ]!={}&]],{n,0,7}]

a(n) = if(n==0, 1, 2^(n-1) - if(n%2==0, (n+2)*2^(n/2-3), (n-1)*2^((n-5)/2))) \\ Andrew Howroyd, Jan 13 2024

a(n) = A011782(n) - A349050(n).

a(n) = 2^(n-1) - (n+2)*2^(n/2-3) for even n > 0; a(n) = 2^(n-1) - (n-1)*2^((n-5)/2) for odd n. - Andrew Howroyd, Jan 13 2024

Terms a(10) and beyond from Andrew Howroyd, Jan 13 2024

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}]

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).

A386585 Triangle read by rows where T(n,k) is the number of integer partitions y of n into k = 0..n parts such that any multiset whose multiplicities are the parts of y is separable.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 1, 1, 0, 0, 1, 2, 2, 1, 1, 0, 0, 1, 3, 3, 2, 1, 1, 0, 0, 1, 3, 4, 3, 2, 1, 1, 0, 0, 1, 5, 5, 5, 3, 2, 1, 1, 0, 0, 1, 4, 7, 6, 5, 3, 2, 1, 1

Offset: 0

Gus Wiseman, Aug 02 2025

We say that such partitions are of separable type.

A multiset is separable iff it has a permutation without any adjacent equal parts.

			Row n = 8 counts the following partitions:
  .  .  44  431  4211  41111  311111  2111111  11111111
            422  3311  32111  221111
            332  3221  22211
                 2222
with the following separable multisets:
  . . 11112222 11112223 11112234 11112345 11123456 11234567 12345678
               11112233 11122234 11122345 11223456
               11122233 11122334 11223345
                        11223344
Triangle begins:
  1
  0  1
  0  0  1
  0  0  1  1
  0  0  1  1  1
  0  0  1  2  1  1
  0  0  1  2  2  1  1
  0  0  1  3  3  2  1  1
  0  0  1  3  4  3  2  1  1
  0  0  1  5  5  5  3  2  1  1
  0  0  1  4  7  6  5  3  2  1  1

This is the separable type case of A072233 or A008284.
Row sums are A336106, ranks A335127.
For separable instead of separable type we have A386583, inseparable A386584.
For inseparable instead of separable we have A386586, sums A025065, ranks A335126.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A239455 counts Look-and-Say partitions, ranks A351294.
A279790 counts disjoint families on strongly normal multisets.
A325534 counts separable multisets, ranks A335433.
A325535 counts inseparable multisets, ranks A335448.
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. A005651, A106351, A111133, A238130, A335434, A386575, A386576, A386579, A386580, A386581, A386582.

sepQ[y_]:=Select[Permutations[y],Length[Split[#]]==Length[y]&]!={};
mst[y_]:=Join@@Table[ConstantArray[k,y[[k]]],{k,Length[y]}];
Table[Length[Select[IntegerPartitions[n,{k}],sepQ[mst[#]]&]],{n,0,5},{k,0,n}]

a(n) = A072233(n) - A386586(n).

A386586 Triangle read by rows where T(n,k) is the number of integer partitions y of n into k parts such that any multiset whose multiplicities are the parts of y is inseparable.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Aug 05 2025

We say that such partitions are of inseparable type. This is different from inseparable partitions (see A386584). A multiset is separable iff it has a permutation without any adjacent equal parts.

			The partition y = (7,2,1) is the multiplicities of the multiset {1,1,1,1,1,1,1,2,2,3}, which is inseparable, so y is counted under T(10,3).
Row n = 10 counts the following partitions (A = 10):
  .  A  91  811  7111  61111  .  .  .  .  .
        82  721  6211
        73  631
        64  622
Triangle begins:
  0
  0 0
  0 1 0
  0 1 0 0
  0 1 1 0 0
  0 1 1 0 0 0
  0 1 2 1 0 0 0
  0 1 2 1 0 0 0 0
  0 1 3 2 1 0 0 0 0
  0 1 3 2 1 0 0 0 0 0
  0 1 4 4 2 1 0 0 0 0 0

This is the inseparable type case of A008284 or A072233.
Row sums shifted left once are A025065 (ranks A335126), separable version A336106 (ranks A335127).
For separable instead of inseparable type we have A386583.
For integer partitions instead of normal multisets we have A386584.
For separable type instead of inseparable type we have A386585.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A239455 counts Look-and-Say partitions, ranks A351294.
A325534 counts separable multisets, ranks A335433.
A325535 counts inseparable multisets, ranks A335448.
A336103 counts normal separable multisets, inseparable A336102.
A351293 counts non-Look-and-Say partitions, ranks A351295.
Cf. A000670, A005651, A106351, A238130, A279790, A386575, A386576, A386579, A386582, A386634.

insepQ[y_]:=Select[Permutations[y],Length[Split[#]]==Length[y]&]=={};
ptm[y_]:=Join@@Table[ConstantArray[k,y[[k]]],{k,Length[y]}];
Table[Length[Select[IntegerPartitions[n,{k}],insepQ[ptm[#]]&]],{n,0,5},{k,0,n}]

a(n) = A072233(n) - A386585(n).

A386634 Number of inseparable type set partitions of {1..n}.

Original entry on oeis.org

0, 0, 1, 1, 5, 6, 37, 50, 345, 502, 3851, 5897, 49854, 79249, 730745, 1195147, 11915997, 19929390, 213332101, 363275555, 4150104224, 7172334477, 87003759195, 152231458128, 1952292972199, 3451893361661, 46625594567852, 83183249675125, 1179506183956655, 2120758970878892

Offset: 0

Gus Wiseman, Aug 09 2025

nonn

A set partition is of inseparable type iff the underlying set has no permutation whose adjacent elements always belong to different blocks. Note that this only depends on the sizes of the blocks.
A set partition is also of inseparable type iff its greatest block size is at least 2 more than the sum of its other block sizes.
This is different from inseparable partitions (A325535) and partitions of inseparable type (A386638 or A025065).

			The a(2) = 1 through a(5) = 6 set partitions:
  {{1,2}}  {{1,2,3}}  {{1,2,3,4}}    {{1,2,3,4,5}}
                      {{1},{2,3,4}}  {{1},{2,3,4,5}}
                      {{1,2,3},{4}}  {{1,2,3,4},{5}}
                      {{1,2,4},{3}}  {{1,2,3,5},{4}}
                      {{1,3,4},{2}}  {{1,2,4,5},{3}}
                                     {{1,3,4,5},{2}}

Alois P. Heinz, Table of n, a(n) for n = 0..250

For separable partitions we have A386583, sums A325534, ranks A335433.
For inseparable partitions we have A386584, sums A325535, ranks A335448.
For separable type partitions we have A386585, sums A336106, ranks A335127.
For inseparable type partitions we have A386586, sums A386638 or A025065, ranks A335126.
The complement is counted by A386633, sums of A386635.
Row sums of A386636.
A000110 counts set partitions, row sums of A048993.
A000670 counts ordered set partitions.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A279790 counts disjoint families on strongly normal multisets.
A335434 counts separable factorizations, inseparable A333487.
A336103 counts normal separable multisets, inseparable A336102.
A386587 counts disjoint families of strict partitions of each prime exponent.
Cf. A001055, A124762, A239455, A335125, A386575, A386579.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
stnseps[stn_]:=Select[Permutations[Union@@stn],And@@Table[Position[stn,#[[i]]][[1,1]]!=Position[stn,#[[i+1]]][[1,1]],{i,Length[#]-1}]&]
Table[Length[Select[sps[Range[n]],stnseps[#]=={}&]],{n,0,5}]

a(12)-a(29) from Alois P. Heinz, Aug 10 2025

A386633 Number of separable type set partitions of {1..n}.

Original entry on oeis.org

1, 1, 1, 4, 10, 46, 166, 827, 3795, 20645, 112124, 672673, 4163743, 27565188, 190168577, 1381763398, 10468226150, 82844940414, 681863474058, 5832378929502, 51720008131148, 474862643822274, 4506628734688128, 44151853623626218, 445956917001833090, 4638586880336637692

Offset: 0

Gus Wiseman, Aug 09 2025

nonn

A set partition is of separable type iff the underlying set has a permutation whose adjacent elements always belong to different blocks. Note that this only depends on the sizes of the blocks.
A set partition is also of separable type iff its greatest block size is at most one more than the sum of all its other block sizes.
This is different from separable partitions (A325534) and partitions of separable type (A336106).

			The a(1) = 1 through a(4) = 10 set partitions:
  {{1}}  {{1},{2}}  {{1},{2,3}}    {{1,2},{3,4}}
                    {{1,2},{3}}    {{1,3},{2,4}}
                    {{1,3},{2}}    {{1,4},{2,3}}
                    {{1},{2},{3}}  {{1},{2},{3,4}}
                                   {{1},{2,3},{4}}
                                   {{1,2},{3},{4}}
                                   {{1},{2,4},{3}}
                                   {{1,3},{2},{4}}
                                   {{1,4},{2},{3}}
                                   {{1},{2},{3},{4}}

Alois P. Heinz, Table of n, a(n) for n = 0..250

For separable partitions see A386583, sums A325534, ranks A335433.
For inseparable partitions see A386584, sums A325535, ranks A335448.
For separable type partitions see A386585, sums A336106, ranks A335127.
For inseparable type partitions see A386586, sums A386638 or A025065, ranks A335126.
The complement is counted by A386634, sums of A386636.
Row sums of A386635.
A000110 counts set partitions, row sums of A048993.
A000670 counts ordered set partitions.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A279790 counts disjoint families on strongly normal multisets.
A335434 counts separable factorizations, inseparable A333487.
A336103 counts normal separable multisets, inseparable A336102.
Cf. A001055, A072233, A106351, A190945, A239455, A239964, A335125, A386575, A386578, A386580, A386587.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
stnseps[stn_]:=Select[Permutations[Union@@stn],And@@Table[Position[stn,#[[i]]][[1,1]]!=Position[stn,#[[i+1]]][[1,1]],{i,Length[#]-1}]&]
Table[Length[Select[sps[Range[n]],stnseps[#]!={}&]],{n,0,5}]

a(12)-a(25) from Alois P. Heinz, Aug 10 2025

A386635 Triangle read by rows where T(n,k) is the number of separable type set partitions of {1..n} into k blocks.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 3, 1, 0, 0, 3, 6, 1, 0, 0, 10, 25, 10, 1, 0, 0, 10, 75, 65, 15, 1, 0, 0, 35, 280, 350, 140, 21, 1, 0, 0, 35, 770, 1645, 1050, 266, 28, 1, 0, 0, 126, 2737, 7686, 6951, 2646, 462, 36, 1, 0, 0, 126, 7455, 32725, 42315, 22827, 5880, 750, 45, 1

Offset: 0

Gus Wiseman, Aug 10 2025

A set partition is of separable type iff the underlying set has a permutation whose adjacent elements always belong to different blocks. Note that this only depends on the sizes of the blocks.
A set partition is also of separable type iff its greatest block size is at most one more than the sum of all its other blocks sizes.
This is different from separable partitions (A325534) and partitions of separable type (A336106).

			Row n = 4 counts the following set partitions:
  .  .  {{1,2},{3,4}}  {{1},{2},{3,4}}  {{1},{2},{3},{4}}
        {{1,3},{2,4}}  {{1},{2,3},{4}}
        {{1,4},{2,3}}  {{1},{2,4},{3}}
                       {{1,2},{3},{4}}
                       {{1,3},{2},{4}}
                       {{1,4},{2},{3}}
Triangle begins:
    1
    0    1
    0    0    1
    0    0    3    1
    0    0    3    6    1
    0    0   10   25   10    1
    0    0   10   75   65   15    1
    0    0   35  280  350  140   21    1

Column k = 2 appears to be A128015.
For separable partitions we have A386583, sums A325534, ranks A335433.
For inseparable partitions we have A386584, sums A325535, ranks A335448.
For separable type partitions we have A386585, sums A336106, ranks A335127.
For inseparable type partitions we have A386586, sums A386638 or A025065, ranks A335126.
Row sums are A386633.
The complement is counted by A386636, row sums A386634.
A000110 counts set partitions, row sums of A048993.
A000670 counts ordered set partitions.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A335434 counts separable factorizations, inseparable A333487.
A336103 counts normal separable multisets, inseparable A336102.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
A386587 counts disjoint families of strict partitions of each prime exponent.
Cf. A072233, A097805, A106351, A116861, A190945, A279790, A333382, A335125, A374252, A386575, A386578.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
stnseps[stn_]:=Select[Permutations[Union@@stn],And@@Table[Position[stn,#[[i]]][[1,1]]!=Position[stn,#[[i+1]]][[1,1]],{i,Length[#]-1}]&];
Table[Length[Select[sps[Range[n]],Length[#]==k&&stnseps[#]!={}&]],{n,0,5},{k,0,n}]

cp's OEIS Frontend

A239455 Number of Look-and-Say partitions of n; see Comments.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A336103 Number of separable multisets of size n covering an initial interval of positive integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A349055 Number of multisets of size n that have an alternating permutation and cover an initial interval of positive integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A386585 Triangle read by rows where T(n,k) is the number of integer partitions y of n into k = 0..n parts such that any multiset whose multiplicities are the parts of y is separable.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A386586 Triangle read by rows where T(n,k) is the number of integer partitions y of n into k parts such that any multiset whose multiplicities are the parts of y is inseparable.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A386634 Number of inseparable type set partitions of {1..n}.

Views

Author

Keywords

Comments