A225485 -id:A225485 - cp's OEIS frontend

A329746 Triangle read by rows where T(n,k) is the number of integer partitions of n > 0 with runs-resistance k, 0 <= k <= n - 1.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 0, 1, 3, 4, 3, 0, 0, 1, 1, 4, 8, 1, 0, 0, 1, 3, 6, 10, 2, 0, 0, 0, 1, 2, 8, 13, 6, 0, 0, 0, 0, 1, 3, 11, 20, 7, 0, 0, 0, 0, 0, 1, 1, 11, 29, 14, 0, 0, 0, 0, 0, 0, 1, 5, 19, 31, 20, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 21 2019

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  1
  1  2  1  1
  1  1  2  3  0
  1  3  4  3  0  0
  1  1  4  8  1  0  0
  1  3  6 10  2  0  0  0
  1  2  8 13  6  0  0  0  0
  1  3 11 20  7  0  0  0  0  0
  1  1 11 29 14  0  0  0  0  0  0
  1  5 19 31 20  1  0  0  0  0  0  0
  1  1 17 50 30  2  0  0  0  0  0  0  0
  1  3 25 64 37  5  0  0  0  0  0  0  0  0
  1  3 29 74 62  7  0  0  0  0  0  0  0  0  0
Row n = 8 counts the following partitions:
  (8)  (44)        (53)    (332)      (4211)
       (2222)      (62)    (422)      (32111)
       (11111111)  (71)    (611)
                   (431)   (3221)
                   (521)   (5111)
                   (3311)  (22211)
                           (41111)
                           (221111)
                           (311111)
                           (2111111)

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Row sums are A000041.
Column k = 1 is A032741.
Column k = 2 is A329745.
A similar invariant is frequency depth; see A323014, A325280.
The version for compositions is A329744.
The version for binary words is A329767.
Cf. A098504, A182850, A225485, A242882, A318928, A325410, A329747.

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

\\ rr(p) gives runs resistance of partition.
rr(p)={my(r=0); while(#p > 1, my(L=List(), k=0); for(i=1, #p, if(i==#p||p[i]<>p[i+1], listput(L, i-k); k=i)); p=Vec(L); r++); r}
row(n)={my(v=vector(n)); forpart(p=n, v[1+rr(Vec(p))]++); v}
{ for(n=1, 10, print(row(n))) } \\ Andrew Howroyd, Jan 19 2023

A225486 Maximal frequency depth for the partitions of n.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 08 2013

nonn

See A225485 for the definition of frequency depth.

The frequency depth of an integer partition is the number of times one must take the multiset of multiplicities to reach (1). For example, the partition (32211) has frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2) -> (1). Differs from A325282 at a(0) and a(1). - Gus Wiseman, Apr 19 2019

			(See A225485.)

Run lengths are A325258, i.e., first differences of Levine's sequence A011784.

Cf. A008284, A116608, A181819, A182850, A182857, A225485, A323014, A323023, A325239, A325242, A325254, A325282, A325283.

c[s_] := c[s] = Select[Table[Count[s, i], {i, 1, Max[s]}], # > 0 &]
f[s_] := f[s] = Drop[FixedPointList[c, s], -2]
t[s_] := t[s] = Length[f[s]]
u[n_] := u[n] = Table[t[Part[IntegerPartitions[n], k]],
    {k, 1, Length[IntegerPartitions[n]]}];
Prepend[Table[Max[u[n]], {n, 2, 10}], 0]
(* second program *)
grw[q_]:=Join@@Table[ConstantArray[i,q[[Length[q]-i+1]]],{i,Length[q]}];
Join@@MapIndexed[ConstantArray[#2[[1]]-1,#1]&,Length[#]-Last[#]&/@NestList[grw,{1,1},6]] (* Gus Wiseman, Apr 19 2019 *)

a(n) = number of terms in row n of the array in A225485, for n > 0.

More terms from Gus Wiseman, Apr 19 2019

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 *)

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}]

A353865 Number of complete rucksack partitions of n. Partitions whose weak run-sums are distinct and cover an initial interval of nonnegative integers.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 5, 2, 3, 4, 3, 2, 4, 3, 3, 4, 4, 3, 4, 3, 4, 5, 5, 4, 6, 4, 6, 5, 4, 5, 6, 5, 6, 7, 6, 5, 9, 6, 6, 7, 6, 8, 9, 6, 6, 8, 9, 7, 9, 9, 7, 10, 9, 8, 13, 7, 10, 11, 8, 9, 10, 11, 12, 9, 11, 9, 15, 12, 12, 19, 13, 16, 16

Offset: 0

Gus Wiseman, Jun 04 2022

nonn

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4). A weak run-sum is the sum of any consecutive constant subsequence.

Do all positive integers appear only finitely many times in this sequence?

			The a(n) compositions for n = 1, 3, 9, 15, 18:
  (1)  (21)   (4311)       (54321)            (543321)
       (111)  (51111)      (532221)           (654111)
              (111111111)  (651111)           (7611111)
                           (81111111)         (111111111111111111)
                           (111111111111111)
For example, the weak runs of y = {7,5,4,4,3,3,3,1,1} are {}, {1}, {1,1}, {3}, {4}, {5}, {3,3}, {7}, {4,4}, {3,3,3}, with sums 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, which are all distinct and cover an initial interval, so y is counted under a(31).

Perfect partitions are counted by A002033, ranked by A325780.
Knapsack partitions are counted by A108917, ranked by A299702.
This is the complete case of A353864, ranked by A353866.
These partitions are ranked by A353867.
A000041 counts partitions, strict A000009.
A275870 counts collapsible partitions, ranked by A300273.
A304442 counts partitions with all equal run-sums, ranked by A353833.
A353832 represents the operation of taking run-sums of a partition.
A353836 counts partitions by number of distinct run-sums.
A353837 counts partitions with distinct run-sums, ranked by A353838.
A353840-A353846 pertain to partition run-sum trajectory.
A353850 counts compositions with all distinct run-sums, ranked by A353852.
A353863 counts partitions whose weak run-sums cover an initial interval.
Cf. A008284, A018818, A225485, A325239, A325862, A353834, A353839.

norqQ[m_]:=Sort[m]==Range[0,Max[m]];
msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
Table[Length[Select[IntegerPartitions[n],norqQ[Total/@Select[msubs[#],SameQ@@#&]]&]],{n,0,15}]

a(n) = my(c=0, s, v); if(n, forpart(p=n, if(p[1]==1, v=List([s=1]); for(i=2, #p, if(p[i]==p[i-1], listput(v, s+=p[i]), listput(v, s=p[i]))); s=#v; listsort(v, 1); if(s==#v&&s==v[s], c++))); c, 1); \\ Jinyuan Wang, Feb 21 2025

More terms from Jinyuan Wang, Feb 21 2025

A351592 Number of Look-and-Say partitions (A239455) of n without distinct multiplicities, i.e., those that are not Wilf partitions (A098859).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 1, 0, 5, 2, 8, 9, 8, 6, 21, 14, 20, 26, 31, 24, 53, 35, 60, 68, 78, 76, 140, 115, 163, 183, 232, 218, 343, 301, 433, 432, 565, 542, 774, 728, 958, 977, 1251, 1220, 1612, 1561, 2053, 2090, 2618, 2609, 3326, 3378

Offset: 0

Gus Wiseman, Feb 16 2022

nonn

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 distinct, so y is counted under A239455(9).
A partition is Wilf iff it has distinct multiplicities of parts. For example, (2,2,2,1,1,1) has multiplicities (3,3), so is not counted under A098859(9).
The Heinz numbers of these partitions are given by A351294 \ A130091.
Is a(17) = 0 the last zero of the sequence?

			The a(9) = 1 through a(18) = 5 partitions are (empty columns not shown):
  n=9:      n=12:       n=15:         n=16:       n=18:
  --------------------------------------------------------------
  (222111)  (333111)    (333222)      (33331111)  (444222)
            (22221111)  (444111)                  (555111)
                        (2222211111)              (3322221111)
                                                  (32222211111)
                                                  (222222111111)

Wilf partitions are counted by A098859, ranked by A130091.
Look-and-Say partitions are counted by A239455, ranked by A351294.
Non-Wilf partitions are counted by A336866, ranked by A130092.
Non-Look-and-Say partitions are counted by A351293, ranked by A351295.
A000569 = number of graphical partitions, complement A339617.
A032020 = number of binary expansions with all distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A225485/A325280 = frequency depth, ranked by A182850/A323014.
A329738 = compositions with all equal run-lengths.
A329739 = compositions with all distinct run-lengths
A351013 = compositions with all distinct runs.
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, A018783, A047966, A181819, A182857, A238130, A305563, A319149, A351203, A351204, A351290.

Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Length/@Split[#]&&Select[Permutations[#], UnsameQ@@Length/@Split[#]&]!={}&]],{n,0,15}]

a(n) = A239455(n) - A098859(n). Here we assume A239455(0) = 1.

More terms from Jinyuan Wang, Feb 14 2025

A325278 Smallest number with adjusted frequency depth n.

Original entry on oeis.org

1, 2, 4, 6, 12, 60, 2520, 1286485200, 35933692027611398678865941374040400000

Offset: 0

Gus Wiseman, Apr 17 2019

nonn

The adjusted frequency depth of a positive integer n is 0 if n = 1, and otherwise it is 1 plus the number of times one must apply A181819 to reach a prime number, where A181819(k = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of k. For example, 180 has adjusted frequency depth 5 because we have: 180 -> 18 -> 6 -> 4 -> 3.

Differs from A182857 in having 2 instead of 3.

A subsequence of A325238.
Cf. A011784, A056239, A112798, A118914, A181819, A181821, A182857, A225485, A323023, A325258, A325272, A325277, A325278, A325280.
Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number).

nn=10000;
fd[n_]:=Switch[n,1,0,?PrimeQ,1,,1+fd[Times@@Prime/@Last/@FactorInteger[n]]];
fds=fd/@Range[nn];
Sort[Table[Position[fds,x][[1,1]],{x,Union[fds]}]]

A325282 Maximum adjusted frequency depth among integer partitions of n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Apr 18 2019

nonn

The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is one plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).
The term "frequency depth" appears to have been coined by Clark Kimberling in A225485 and A225486, and can be applied to both integers (A323014) and integer partitions (A325280).
Run lengths are A325258, i.e., first differences of Levine's sequence A011784 (except at n = 1).

Cf. A011784, A032741, A127002, A181819, A225486, A275870, A323014, A323023, A325245, A325254, A325258, A325278, A325282, A325283.

Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
Table[Max@@fdadj/@IntegerPartitions[n],{n,0,30}]

a(0) = 0; a(1) = 1; a(n > 1) = A225486(n).

A378622 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the strict partition numbers A000009.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 2, 1, 1, 1, 2, 0, -1, -2, -3, 3, 1, 1, 2, 4, 7, 4, 1, 0, -1, -3, -7, -14, 5, 1, 0, 0, 1, 4, 11, 25, 6, 1, 0, 0, 0, -1, -5, -16, -41, 8, 2, 1, 1, 1, 1, 2, 7, 23, 64, 10, 2, 0, -1, -2, -3, -4, -6, -13, -36, -100, 12, 2, 0, 0, 1, 3, 6, 10, 16, 29, 65, 165

Offset: 0

Gus Wiseman, Dec 13 2024

			As a table (read by antidiagonals downward):
        n=0:  n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:
  ----------------------------------------------------------
  k=0:   1     1     1     2     2     3     4     5     6
  k=1:   0     0     1     0     1     1     1     1     2
  k=2:   0     1    -1     1     0     0     0     1     0
  k=3:   1    -2     2    -1     0     0     1    -1     0
  k=4:  -3     4    -3     1     0     1    -2     1     1
  k=5:   7    -7     4    -1     1    -3     3     0    -3
  k=6: -14    11    -5     2    -4     6    -3    -3     7
  k=7:  25   -16     7    -6    10    -9     0    10   -14
  k=8: -41    23   -13    16   -19     9    10   -24    24
  k=9:  64   -36    29   -35    28     1   -34    48   -34
As a triangle (read by rows):
   1
   1   0
   1   0   0
   2   1   1   1
   2   0  -1  -2  -3
   3   1   1   2   4   7
   4   1   0  -1  -3  -7 -14
   5   1   0   0   1   4  11  25
   6   1   0   0   0  -1  -5 -16 -41
   8   2   1   1   1   1   2   7  23  64

Rows are: A000009 (k=0), A087897 (k=1, without first term), A378972 (k=2).
For primes we have A095195 or A376682.
For partitions we have A175804.
First column is A293467 (up to sign).
For composites we have A377033.
For squarefree numbers we have A377038.
For nonsquarefree numbers we have A377046.
For prime powers we have A377051.
Position of first zero in each row is A377285.
Triangle's row-sums are A378970, absolute A378971.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A047966, A098859, A225486, A325244, A325282.
Cf. A008284, A116608, A325242, A225485 or A325280.

nn=20;
t=Table[Take[Differences[PartitionsQ/@Range[0,2nn],k],nn],{k,0,nn}];
Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

A325245 Number of integer partitions of n with adjusted frequency depth 3.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 4, 6, 8, 11, 11, 19, 17, 25, 29, 37, 37, 56, 53, 75, 80, 99, 103, 145, 143, 181, 199, 247, 255, 336, 339, 426, 459, 548, 590, 738, 759, 916, 999, 1192, 1259, 1529, 1609, 1915, 2083, 2406, 2589, 3085, 3267, 3809, 4134, 4763, 5119, 5964

Offset: 0

Gus Wiseman, Apr 15 2019

nonn

The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2). The enumeration of integer partitions by adjusted frequency depth is given by A325280. The adjusted frequency depth of the integer partition with Heinz number n is given by A323014.

			The a(3) = 1 through a(10) = 11 partitions:
  (21)  (31)  (32)  (42)    (43)   (53)    (54)      (64)
              (41)  (51)    (52)   (62)    (63)      (73)
                    (321)   (61)   (71)    (72)      (82)
                    (2211)  (421)  (431)   (81)      (91)
                                   (521)   (432)     (532)
                                   (3311)  (531)     (541)
                                           (621)     (631)
                                           (222111)  (721)
                                                     (3322)
                                                     (4321)
                                                     (4411)

Column k = 3 of A225485 and A325280.

Cf. A008284, A047966, A116608, A127002, A181819, A182850, A323014, A323023, A325239, A325246, A325254, A325268, A325280.

fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
Table[Length[Select[IntegerPartitions[n],fdadj[#]==3&]],{n,0,30}]

cp's OEIS Frontend

A329746 Triangle read by rows where T(n,k) is the number of integer partitions of n > 0 with runs-resistance k, 0 <= k <= n - 1.

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A225486 Maximal frequency depth for the partitions 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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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.

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A353865 Number of complete rucksack partitions of n. Partitions whose weak run-sums are distinct and cover an initial interval of nonnegative integers.

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A351592 Number of Look-and-Say partitions (A239455) of n without distinct multiplicities, i.e., those that are not Wilf partitions (A098859).

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A325278 Smallest number with adjusted frequency depth n.

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A325282 Maximum adjusted frequency depth among integer partitions of n.

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