A329746 -id:A329746 - 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 *)

A351293 Number of non-Look-and-Say partitions of n. Number of integer partitions of n such that there is no way to choose a disjoint strict integer partition of each multiplicity.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 9, 14, 21, 28, 44, 56, 80, 111, 148, 192, 264, 335, 447, 575, 743, 937, 1213, 1513, 1924, 2396, 3011, 3715, 4646, 5687, 7040, 8600, 10556, 12804, 15650, 18897, 22930, 27593, 33296, 39884, 47921, 57168, 68360, 81295, 96807, 114685

Offset: 0

Gus Wiseman, Feb 16 2022

nonn

First differs from A336866 (non-Wilf partitions) at a(9) = 14, A336866(9) = 15, the difference being the partition (2,2,2,1,1,1).

See A239455 for the definition of Look-and-Say partitions.

			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)
                                             (321111)

The complement is counted by A239455, ranked by A351294.
These are all non-Wilf partitions (counted by A336866, ranked by A130092).
A variant for runs is A351203, complement A351204, ranked by A351201.
These partitions appear to be ranked by A351295.
Non-Wilf partitions in the complement are counted by A351592.
A000569 = graphical partitions, complement A339617.
A032020 = number of binary expansions with all distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A098859 = Wilf partitions (distinct multiplicities), ranked by A130091.
A181819 = Heinz number of the prime signature of n (prime shadow).
A329738 = compositions with all equal run-lengths.
A329739 = compositions with all distinct run-lengths, for all runs A351013.
A351017 = binary words with all distinct run-lengths, for all runs A351016.
A351292 = patterns with all distinct run-lengths, for all runs A351200.
Cf. A000041, A008284, A047966, A182857, A225485, A238130, A297770, A304660, A305563, A329740, A329746, A351202, A351291.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
Table[Length[Select[IntegerPartitions[n],Length[disjointFamilies[#]]==0&]],{n,0,15}] (* Gus Wiseman, Aug 13 2025 *)

a(n) = A000041(n) - A239455(n).

Edited by Gus Wiseman, Aug 12 2025

A351295 Numbers whose multiset of prime factors has no permutation with all distinct run-lengths.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126, 129, 130, 132, 133, 134, 138, 140

Offset: 1

Gus Wiseman, Feb 16 2022

nonn

First differs from A130092 (non-Wilf partitions) in lacking 216.

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 terms together with their prime indices begin:
      6: (2,1)         46: (9,1)         84: (4,2,1,1)
     10: (3,1)         51: (7,2)         85: (7,3)
     14: (4,1)         55: (5,3)         86: (14,1)
     15: (3,2)         57: (8,2)         87: (10,2)
     21: (4,2)         58: (10,1)        90: (3,2,2,1)
     22: (5,1)         60: (3,2,1,1)     91: (6,4)
     26: (6,1)         62: (11,1)        93: (11,2)
     30: (3,2,1)       65: (6,3)         94: (15,1)
     33: (5,2)         66: (5,2,1)       95: (8,3)
     34: (7,1)         69: (9,2)        100: (3,3,1,1)
     35: (4,3)         70: (4,3,1)      102: (7,2,1)
     36: (2,2,1,1)     74: (12,1)       105: (4,3,2)
     38: (8,1)         77: (5,4)        106: (16,1)
     39: (6,2)         78: (6,2,1)      110: (5,3,1)
     42: (4,2,1)       82: (13,1)       111: (12,2)
For example, the prime indices of 150 are {1,2,3,3}, with permutations and run-lengths (right):
  (3,3,2,1) -> (2,1,1)
  (3,3,1,2) -> (2,1,1)
  (3,2,3,1) -> (1,1,1,1)
  (3,2,1,3) -> (1,1,1,1)
  (3,1,3,2) -> (1,1,1,1)
  (3,1,2,3) -> (1,1,1,1)
  (2,3,3,1) -> (1,2,1)
  (2,3,1,3) -> (1,1,1,1)
  (2,1,3,3) -> (1,1,2)
  (1,3,3,2) -> (1,2,1)
  (1,3,2,3) -> (1,1,1,1)
  (1,2,3,3) -> (1,1,2)
Since none have all distinct run-lengths, 150 is in the sequence.

Wilf partitions are counted by A098859, ranked by A130091.
Non-Wilf partitions are counted by A336866, ranked by A130092.
A variant for runs is A351201, counted by A351203 (complement A351204).
These partitions appear to be counted by A351293.
The complement is A351294, apparently counted by A239455.
A032020 = number of binary expansions with distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A056239 = sum of prime indices, row sums of A112798.
A165413 = number of distinct run-lengths in binary expansion.
A181819 = Heinz number of prime signature (prime shadow).
A182850/A323014 = frequency depth, counted by A225485/A325280.
A297770 = number of distinct runs in binary expansion.
A320922 ranks graphical partitions, complement A339618, counted by A000569.
A329739 = compositions with all distinct run-lengths, for all runs A351013.
A329747 = runs-resistance, counted by A329746.
A333489 ranks anti-runs, complement A348612.
A351017 = binary words with all distinct run-lengths, for all runs A351016.
Cf. A000961, A001221, A001222, A175413, A182857, A304660, A320924, A328592, A351202, A351290.

Select[Range[100],Select[Permutations[Join@@ ConstantArray@@@FactorInteger[#]],UnsameQ@@Length/@Split[#]&]=={}&]

Name edited by Gus Wiseman, Aug 13 2025

A329744 Triangle read by rows where T(n,k) is the number of compositions of n > 0 with runs-resistance k, 0 <= k <= n - 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 3, 2, 1, 1, 6, 6, 2, 1, 3, 15, 9, 4, 0, 1, 1, 22, 22, 16, 2, 0, 1, 3, 41, 38, 37, 8, 0, 0, 1, 2, 72, 69, 86, 26, 0, 0, 0, 1, 3, 129, 124, 175, 78, 2, 0, 0, 0, 1, 1, 213, 226, 367, 202, 14, 0, 0, 0, 0, 1, 5, 395, 376, 750, 469, 52, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 21 2019

A composition of n is a finite sequence of positive integers with sum n.

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.

			Triangle begins:
   1
   1   1
   1   1   2
   1   2   3   2
   1   1   6   6   2
   1   3  15   9   4   0
   1   1  22  22  16   2   0
   1   3  41  38  37   8   0   0
   1   2  72  69  86  26   0   0   0
   1   3 129 124 175  78   2   0   0   0
   1   1 213 226 367 202  14   0   0   0   0
   1   5 395 376 750 469  52   0   0   0   0   0
Row n = 6 counts the following compositions:
  (6)  (33)      (15)    (114)    (1131)
       (222)     (24)    (411)    (1311)
       (111111)  (42)    (1113)   (11121)
                 (51)    (1221)   (12111)
                 (123)   (2112)
                 (132)   (3111)
                 (141)   (11112)
                 (213)   (11211)
                 (231)   (21111)
                 (312)
                 (321)
                 (1122)
                 (1212)
                 (2121)
                 (2211)

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Row sums are A000079.
Column k = 1 is A032741.
Column k = 2 is A329745.
Column k = n - 2 is A329743.
The version for partitions is A329746.
The version with rows reversed is A329750.
Cf. A000740, A008965, A098504, A242882, A318928, A329747.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==k&]],{n,10},{k,0,n-1}]

A329747 Runs-resistance of the sequence of prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 21 2019

nonn

First differs from A304455 at a(90) = 3, A304455(90) = 4.
For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.
A prime index of n is a number m such that prime(m) divides n. The sequence of prime indices of n is row n of A112798.

			We have (1,2,2,3) -> (1,2,1) -> (1,1,1) -> (3), so a(90) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.
Index entries for sequences related to prime indices in the factorization of n.

The version for partitions is A329746.
The version for compositions is A329744.
The version for binary words is A329767.
The version for binary expansion is A318928.
Cf. A008578 (positions of 0's), A056239, A112798, A329745, A329750.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
Table[runsres[primeMS[n]],{n,50}]

pis_to_runs(n) = { my(runs=List([]), f=factor(n)); for(i=1,#f~,while(f[i,2], listput(runs,primepi(f[i,1])); f[i,2]--)); (runs); };
runlengths(lista) = if(!#lista, lista, if(1==#lista, List([1]), my(runs=List([]), rl=1); for(i=1, #lista, if((i< #lista) && (lista[i]==lista[i+1]), rl++, listput(runs,rl); rl=1)); (runs)));
A329747(n) = { my(runs=pis_to_runs(n)); for(i=0,oo,if(#runs<=1, return(i), runs = runlengths(runs))); }; \\ Antti Karttunen, Jan 20 2025

More terms from Antti Karttunen, Jan 20 2025

A319411 Triangle read by rows: T(n,k) = number of binary vectors of length n with runs-resistance k (1 <= k <= n).

Original entry on oeis.org

2, 2, 2, 2, 2, 4, 2, 4, 6, 4, 2, 2, 12, 12, 4, 2, 6, 30, 18, 8, 0, 2, 2, 44, 44, 32, 4, 0, 2, 6, 82, 76, 74, 16, 0, 0, 2, 4, 144, 138, 172, 52, 0, 0, 0, 2, 6, 258, 248, 350, 156, 4, 0, 0, 0, 2, 2, 426, 452, 734, 404, 28, 0, 0, 0, 0, 2, 10, 790, 752, 1500, 938, 104, 0, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane, Sep 20 2018

"Runs-resistance" is defined in A318928.
Row sums are 2,4,8,16,... (the binary vectors may begin with 0 or 1).
This is similar to A329767 but without the k = 0 column and with a different row n = 1. - Gus Wiseman, Nov 25 2019

			Triangle begins:
2,
2, 2,
2, 2, 4,
2, 4, 6, 4,
2, 2, 12, 12, 4,
2, 6, 30, 18, 8, 0,
2, 2, 44, 44, 32, 4, 0,
2, 6, 82, 76, 74, 16, 0, 0,
2, 4, 144, 138, 172, 52, 0, 0, 0,
2, 6, 258, 248, 350, 156, 4, 0, 0, 0,
2, 2, 426, 452, 734, 404, 28, 0, 0, 0, 0,
2, 10, 790, 752, 1500, 938, 104, 0, 0, 0, 0, 0,
...
Lenormand gives the first 20 rows.
The calculation of row 4 is as follows.
We may assume the first bit is a 0, and then double the answers.
vector / runs / steps to reach a single number:
0000 / 4 / 1
0001 / 31 -> 11 -> 2 / 3
0010 / 211 -> 12 -> 11 -> 2 / 4
0011 / 22 -> 2 / 2
0100 / 112 -> 21 -> 11 -> 2 / 4
0101 / 1111 -> 4 / 2
0110 / 121 -> 111 -> 3 / 3
0111 / 13 -> 11 -> 2 / 3
and we get 1 (once), 2 (twice), 3 (three times) and 4 (twice).
So row 4 is: 2,4,6,4.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..1830
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003.

Row sums are A000079.
Column k = 2 is 2 * A032741 = A319410.
Column k = 3 is 2 * A329745 (because runs-resistance 2 for compositions corresponds to runs-resistance 3 for binary words).
The version for compositions is A329744.
The version for partitions is A329746.
The number of nonzero entries in row n > 0 is A319412(n).
The runs-resistance of the binary expansion of n is A318928.
Cf. A096365, A225485, A245563, A319413, A319414, A319415, A325280, A329747, A329750, A329767, A329870.

runsresist[q_]:=If[Length[q]==1,1,Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1];
Table[Length[Select[Tuples[{0,1},n],runsresist[#]==k&]],{n,10},{k,n}] (* Gus Wiseman, Nov 25 2019 *)

A332833 Number of compositions of n whose run-lengths are neither weakly increasing nor weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 3, 8, 27, 75, 185, 441, 1025, 2276, 4985, 10753, 22863, 48142, 100583, 208663, 430563, 884407, 1809546, 3690632, 7506774, 15233198, 30851271, 62377004, 125934437, 253936064, 511491634, 1029318958, 2069728850, 4158873540, 8351730223, 16762945432

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(6) = 3 and a(7) = 8 compositions:
  (1221)   (2113)
  (2112)   (3112)
  (11211)  (11311)
           (12112)
           (21112)
           (21121)
           (111211)
           (112111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Unimodal Sequence.

The case of partitions is A332641.
The version for unsorted prime signature is A332831.
The version for the compositions themselves (not run-lengths) is A332834.
The complement is counted by A332835.
Unimodal compositions are A001523.
Partitions with weakly increasing run-lengths are A100883.
Compositions that are not unimodal are A115981.
Compositions with equal run-lengths are A329738.
Compositions whose run-lengths are unimodal are A332726.
Compositions whose run-lengths are not unimodal are A332727.
Partitions with weakly increasing or weakly decreasing run-lengths: A332745.
Compositions with weakly increasing run-lengths are A332836.
Compositions that are neither unimodal nor is their negation are A332870.
Cf. A001462, A072704, A072706, A107429, A181819, A329398, A329744, A329746, A329766, A332273, A332640, A332746.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!Or[LessEqual@@Length/@Split[#],GreaterEqual@@Length/@Split[#]]&]],{n,0,10}]

a(n) = 2^(n - 1) - 2 * A332836(n) + A329738(n).

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020

A329745 Number of compositions of n with runs-resistance 2.

Original entry on oeis.org

0, 0, 2, 3, 6, 15, 22, 41, 72, 129, 213, 395, 660, 1173, 2031, 3582, 6188, 10927, 18977, 33333, 58153, 101954, 178044, 312080, 545475, 955317, 1670990, 2925711, 5118558, 8960938, 15680072, 27447344, 48033498, 84076139, 147142492, 257546234, 450748482, 788937188

Offset: 1

Gus Wiseman, Nov 21 2019

nonn

A composition of n is a finite sequence of positive integers with sum n.
For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.
These are non-constant compositions with equal run-lengths (A329738).

			The a(3) = 2 through a(6) = 15 compositions:
  (1,2)  (1,3)    (1,4)    (1,5)
  (2,1)  (3,1)    (2,3)    (2,4)
         (1,2,1)  (3,2)    (4,2)
                  (4,1)    (5,1)
                  (1,3,1)  (1,2,3)
                  (2,1,2)  (1,3,2)
                           (1,4,1)
                           (2,1,3)
                           (2,3,1)
                           (3,1,2)
                           (3,2,1)
                           (1,1,2,2)
                           (1,2,1,2)
                           (2,1,2,1)
                           (2,2,1,1)

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Column k = 2 of A329744.
Column k = n - 2 of A329750.
Cf. A000740, A003242, A008965, A098504, A242882, A318928, A329743, A329746, A329747, A329767.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==2&]],{n,10}]

seq(n)={my(b=Vec(1/(1 - sum(k=1, n, x^k/(1+x^k) + O(x*x^n)))-1)); vector(n, k, sumdiv(k, d, b[d]-1))} \\ Andrew Howroyd, Dec 30 2020

a(n) = A329738(n) - A000005(n).

a(n) = Sum_{d|n} (A003242(d) - 1). - Andrew Howroyd, Dec 30 2020

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020

A329767 Triangle read by rows where T(n,k) is the number of binary words of length n >= 0 with runs-resistance k, 0 <= k <= n.

Original entry on oeis.org

1, 2, 0, 0, 2, 2, 0, 2, 2, 4, 0, 2, 4, 6, 4, 0, 2, 2, 12, 12, 4, 0, 2, 6, 30, 18, 8, 0, 0, 2, 2, 44, 44, 32, 4, 0, 0, 2, 6, 82, 76, 74, 16, 0, 0, 0, 2, 4, 144, 138, 172, 52, 0, 0, 0, 0, 2, 6, 258, 248, 350, 156, 4, 0, 0, 0, 0, 2, 2, 426, 452, 734, 404, 28, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Nov 21 2019

A composition of n is a finite sequence of positive integers with sum n.
For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.
Except for the k = 0 column and the n = 0 and n = 1 rows, this is the triangle appearing on page 3 of Lenormand, which is A319411. Unlike A318928, we do not here require that a(n) >= 1.
The n = 0 row is chosen to ensure that the row-sums are A000079, although the empty word arguably has indeterminate runs-resistance.

			Triangle begins:
   1
   2   0
   0   2   2
   0   2   2   4
   0   2   4   6   4
   0   2   2  12  12   4
   0   2   6  30  18   8   0
   0   2   2  44  44  32   4   0
   0   2   6  82  76  74  16   0   0
   0   2   4 144 138 172  52   0   0   0
   0   2   6 258 248 350 156   4   0   0   0
   0   2   2 426 452 734 404  28   0   0   0   0
For example, row n = 4 counts the following words:
  0000  0011  0001  0010
  1111  0101  0110  0100
        1010  0111  1011
        1100  1000  1101
              1001
              1110

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Row sums are A000079.
Column k = 2 is A319410.
Column k = 3 is 2 * A329745.
The version for compositions is A329744.
The version for partitions is A329746.
The number of nonzero entries in row n > 0 is A319412(n).
The runs-resistance of the binary expansion of n is A318928.
Cf. A001037, A096365, A225485, A245563, A319411, A325280, A329747, A329750.

runsres[q_]:=If[Length[q]==1,0,Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1];
Table[Length[Select[Tuples[{0,1},n],runsres[#]==k&]],{n,0,10},{k,0,n}]

A332726 Number of compositions of n whose run-lengths are unimodal.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 31, 61, 120, 228, 438, 836, 1580, 2976, 5596, 10440, 19444, 36099, 66784, 123215, 226846, 416502, 763255, 1395952, 2548444, 4644578, 8452200, 15358445, 27871024, 50514295, 91446810, 165365589, 298730375, 539127705, 972099072, 1751284617, 3152475368

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

A composition of n is a finite sequence of positive integers summing to n.

			The only composition of 6 whose run-lengths are not unimodal is (1,1,2,1,1).

Andrew Howroyd, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Unimodal Sequence.

Looking at the composition itself (not run-lengths) gives A001523.
The case of partitions is A332280, with complement counted by A332281.
The complement is counted by A332727.
Unimodal compositions are A001523.
Unimodal normal sequences appear to be A007052.
Non-unimodal compositions are A115981.
Compositions with normal run-lengths are A329766.
Numbers whose prime signature is not unimodal are A332282.
Partitions whose 0-appended first differences are unimodal are A332283, with complement A332284, with Heinz numbers A332287.
Compositions whose negated run-lengths are unimodal are A332578.
Compositions whose negated run-lengths are not unimodal are A332669.
Compositions whose run-lengths are weakly increasing are A332836.
Cf. A072706, A100883, A181819, A227038, A328509, A329744, A329746, A332642, A332670, A332741, A332833, A332835.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],unimodQ[Length/@Split[#]]&]],{n,0,10}]

step(M, m)={my(n=matsize(M)[1]); for(p=m+1, n, my(v=vector((p-1)\m, i, M[p-i*m,i]), s=vecsum(v)); M[p,]+=vector(#M,i,s-if(i<=#v, v[i]))); M}
desc(M, m)={my(n=matsize(M)[1]); while(m>1, m--; M=step(M,m)); vector(n, i, vecsum(M[i,]))/(#M-1)}
seq(n)={my(M=matrix(n+1, n+1, i, j, i==1), S=M[,1]~); for(m=1, n, my(D=M); M=step(M, m); D=(M-D)[m+1..n+1,1..n-m+2]; S+=concat(vector(m), desc(D,m))); S} \\ Andrew Howroyd, Dec 31 2020

a(n) + A332727(n) = 2^(n - 1).

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2020

cp's OEIS Frontend

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

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A351293 Number of non-Look-and-Say partitions of n. Number of integer partitions of n such that there is no way to choose a disjoint strict integer partition of each multiplicity.

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A351295 Numbers whose multiset of prime factors has no permutation with all distinct run-lengths.

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A329744 Triangle read by rows where T(n,k) is the number of compositions of n > 0 with runs-resistance k, 0 <= k <= n - 1.

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A329747 Runs-resistance of the sequence of prime indices of n.

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A319411 Triangle read by rows: T(n,k) = number of binary vectors of length n with runs-resistance k (1 <= k <= n).

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A332833 Number of compositions of n whose run-lengths are neither weakly increasing nor weakly decreasing.

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A329745 Number of compositions of n with runs-resistance 2.

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