A325280 -id:A325280 - cp's OEIS frontend

A325254 Number of integer partitions of n with the maximum adjusted frequency depth for partitions of n.

0, 1, 1, 1, 1, 3, 3, 1, 3, 7, 10, 17, 27, 38, 1, 4, 8, 17, 31, 52, 83, 122, 181, 257, 361, 499, 684, 910, 1211, 1595, 2060, 2663, 3406, 4315, 5426, 6784, 8417, 10466, 12824, 15721, 19104, 23267, 1, 5, 14, 36, 76, 143, 269, 446, 738, 1143, 1754, 2570, 3742, 5269

Offset: 0

Gus Wiseman, Apr 16 2019

nonn

The Heinz numbers of these partitions are given by A325283.
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 maximum adjusted frequency depth for integer partitions of n is given by A325282.
Essentially, the last numbers of rows of the array in A225485. - Clark Kimberling, Sep 13 2022

			The a(1) = 1 through a(11) = 17 partitions:
  1  11  21  211  221   411    3211  3221   3321    5221     4322
                  311   3111         4211   4221    5311     4331
                  2111  21111        32111  4311    6211     4421
                                            5211    32221    5411
                                            32211   33211    6221
                                            42111   42211    6311
                                            321111  43111    7211
                                                    52111    33221
                                                    421111   42221
                                                    3211111  43211
                                                             52211
                                                             53111
                                                             62111
                                                             431111
                                                             521111
                                                             4211111
                                                             32111111

Cf. A011784, A181819, A182850, A182857, A225486, A323014, A323023, A325246, A325258, A325278, A325281, 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).

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

A377285 Position of first 0 in the n-th differences of the strict partition numbers A000009, or 0 if 0 does not appear.

Original entry on oeis.org

0, 1, 1, 5, 5, 8, 20, 7, 22

Offset: 0

Gus Wiseman, Dec 12 2024

Open problem: Do the 9th differences of the strict integer partition numbers contain a zero? If so, we must have a(9) > 10^5.

a(12) = 47. Conjecture: a(n) = 0 for n > 12. - Chai Wah Wu, Dec 15 2024

			The 7th differences of A000009 are: 25, -16, 7, -6, 10, -9, 0, 10, ... so a(7) = 7.

For primes we have A376678.
For composites we have A377037.
For squarefree numbers we have A377042.
For nonsquarefree numbers we have A377050.
For prime-powers we have A377055.
Position of first zero in each row of A378622. See also:
- A175804 is the version for partitions.
- A293467 gives first column (up to sign).
- A378970 gives row-sums.
- A378971 gives row-sums of absolute value.
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, A325268, A225485 or A325280.

Table[Position[Differences[PartitionsQ/@Range[0,100],k],0][[1,1]],{k,1,8}]

a(n, nn=100) = my(q='q+O('q^nn), v=Vec(eta(q^2)/eta(q))); for (i=1, n, my(w=vector(#v-1, k, v[k+1]-v[k])); v = w;); my(vz=select(x->x==0, v, 1)); if (#vz, vz[1]); \\ Michel Marcus, Dec 15 2024

A353389 Create the sequence of all positive integers > 1 that are prime or whose prime shadow (A181819) is a divisor that is already in the sequence. Then remove all the primes.

Original entry on oeis.org

9, 36, 125, 225, 441, 1089, 1260, 1521, 1980, 2340, 2401, 2601, 2772, 3060, 3249, 3276, 3420, 4140, 4284, 4761, 4788, 5148, 5220, 5580, 5796, 6660, 6732, 7308, 7380, 7524, 7569, 7740, 7812, 7956, 8460, 8649, 8892, 9108, 9324, 9540, 10332, 10620, 10764, 10836

Offset: 1

Gus Wiseman, May 15 2022

nonn

We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

Said differently, these are nonprime numbers > 1 whose prime shadow is a divisor that is either a prime number or a number already in the sequence.

			The initial terms and their prime indices:
     9: {2,2}
    36: {1,1,2,2}
   125: {3,3,3}
   225: {2,2,3,3}
   441: {2,2,4,4}
  1089: {2,2,5,5}
  1260: {1,1,2,2,3,4}
  1521: {2,2,6,6}
  1980: {1,1,2,2,3,5}

The first term that is not a perfect power A001597 is 1260.
Without the recursion we have A325755 (a superset), counted by A325702.
Before removing the primes we had A353393.
These partitions are counted by A353426 minus one.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A182850 and A323014 give frequency depth, counted by A225485 and A325280.
A325131 lists numbers relatively prime to their prime shadow.
Cf. A000005, A000040, A005117, A316413, A316428, A324850, A353394, A353395, A353397, A353399.

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
suQ[n_]:=PrimeQ[n]||Divisible[n,red[n]]&&suQ[red[n]];
Select[Range[2,2000],suQ[#]&&!PrimeQ[#]&]

A182863 Members m of A025487 such that, if k appears in m's prime signature, k-1 appears at least as often as k (for any integer k > 1).

Original entry on oeis.org

1, 2, 6, 12, 30, 60, 210, 360, 420, 1260, 2310, 2520, 4620, 13860, 27720, 30030, 60060, 75600, 138600, 180180, 360360, 510510, 831600, 900900, 1021020, 1801800, 3063060, 6126120, 9699690, 10810800, 15315300, 19399380, 30630600, 37837800

Offset: 1

Matthew Vandermast, Jan 14 2011

nonn

Members m of A025487 such that A181819(m) is also a member of A025487.
If prime signatures are considered as partitions, these are the members of A025487 whose prime signature is conjugate to the prime signature of a member of A181818.
Also the least number with each sorted prime metasignature, where a number's metasignature is the sequence of multiplicities of exponents in its prime factorization. For example, 2520 has prime indices {1,1,1,2,2,3,4}, sorted prime signature {1,1,2,3}, and sorted prime metasignature {1,1,2}. - Gus Wiseman, May 21 2022

			The prime signature of 360360 = 2^3*3^2*5*7*11*13 is (3,2,1,1,1,1). 2 appears as many times as 3 in 360360's prime signature, and 1 appears more times than 2. Since 360360 is also a member of A025487, it is a member of this sequence.
From _Gus Wiseman_, May 21 2022: (Start)
The terms together with their sorted prime signatures and sorted prime metasignatures begin:
      1: {}                -> {}            -> {}
      2: {1}               -> {1}           -> {1}
      6: {1,2}             -> {1,1}         -> {2}
     12: {1,1,2}           -> {1,2}         -> {1,1}
     30: {1,2,3}           -> {1,1,1}       -> {3}
     60: {1,1,2,3}         -> {1,1,2}       -> {1,2}
    210: {1,2,3,4}         -> {1,1,1,1}     -> {4}
    360: {1,1,1,2,2,3}     -> {1,2,3}       -> {1,1,1}
    420: {1,1,2,3,4}       -> {1,1,1,2}     -> {1,3}
   1260: {1,1,2,2,3,4}     -> {1,1,2,2}     -> {2,2}
   2310: {1,2,3,4,5}       -> {1,1,1,1,1}   -> {5}
   2520: {1,1,1,2,2,3,4}   -> {1,1,2,3}     -> {1,1,2}
   4620: {1,1,2,3,4,5}     -> {1,1,1,1,2}   -> {1,4}
  13860: {1,1,2,2,3,4,5}   -> {1,1,1,2,2}   -> {2,3}
  27720: {1,1,1,2,2,3,4,5} -> {1,1,1,2,3}   -> {1,1,3}
  30030: {1,2,3,4,5,6}     -> {1,1,1,1,1,1} -> {6}
  60060: {1,1,2,3,4,5,6}   -> {1,1,1,1,1,2} -> {1,5}
(End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1444 terms from Amiram Eldar)
Eric Weisstein's World of Mathematics, Conjugate Partition.

Intersection of A025487 and A179983.
Subsequence of A129912 and A181826.
Includes all members of A182862.
Positions of first appearances in A353742, unordered version A238747.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A005361 gives product of prime signature, firsts A353500 (sorted A085629).
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
A182850 gives frequency depth of prime indices, counted by A225485.
A323014 gives adjusted frequency depth of prime indices, counted by A325280.
Cf. A000040, A055932, A070175, A097318, A304678, A325238, A353507, A353745.

nn=1000;
r=Table[Sort[Length/@Split[Sort[Last/@If[n==1,{},FactorInteger[n]]]]],{n,nn}];
Select[Range[nn],!MemberQ[Take[r,#-1],r[[#]]]&] (* Gus Wiseman, May 21 2022 *)

A325246 Number of integer partitions of n with adjusted frequency depth equal to their length.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 4, 4, 6, 8, 14, 15, 21, 26, 34, 42, 51, 60, 74, 86, 102, 117, 137, 155, 178, 202, 228, 255, 286, 317, 355, 390, 430, 472, 519, 566, 617, 670, 728, 787, 852, 916, 988, 1060, 1137, 1218, 1303, 1389, 1482, 1577, 1679, 1781, 1890, 2001, 2120

Offset: 0

Gus Wiseman, Apr 15 2019

nonn

The Heinz numbers of these partitions are given by A325266.

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(1) = 1 through a(10) = 14 partitions (A = 10):
  (1)  (2)   (3)  (4)   (5)     (6)     (7)     (8)      (9)      (A)
       (11)       (22)  (2111)  (33)    (421)   (44)     (432)    (55)
                                (321)   (2221)  (431)    (531)    (532)
                                (3111)  (4111)  (521)    (621)    (541)
                                                (5111)   (3222)   (631)
                                                (32111)  (6111)   (721)
                                                         (32211)  (3331)
                                                         (42111)  (4222)
                                                                  (7111)
                                                                  (32221)
                                                                  (33211)
                                                                  (42211)
                                                                  (43111)
                                                                  (52111)

Cf. A001222, A008284, A127002, A181819, A182850, A225485, A323014, A323023, A325254, A325266, A325277, A325280, A325281.

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

A325250 Number of integer partitions of n whose omega-sequence is strict (no repeated parts).

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 5, 2, 5, 4, 6, 2, 11, 3, 10, 12, 17, 12, 31, 22, 42, 47, 57, 60, 98, 94, 119, 143, 174, 182, 256, 253, 321, 365, 425, 480, 615, 645, 803, 946, 1180, 1341, 1766, 2021, 2607, 3145, 3951, 4727, 6123, 7236, 9136

Offset: 0

Gus Wiseman, Apr 16 2019

nonn

The omega-sequence of an integer partition is the sequence of lengths of the multisets obtained by repeatedly taking the multiset of multiplicities until a singleton is reached. For example, the partition (32211) has chain of multisets of multiplicities {1,1,2,2,3} -> {1,2,2} -> {1,2} -> {1,1} -> {2}, so its omega-sequence is (5,3,2,2,1).

The Heinz numbers of these partitions are given by A325247.

			The a(1) = 1 through a(10) = 6 partitions (A = 10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        333        55
              1111         222              2222      222111     3322
                           2211             3311      111111111  4411
                           111111           11111111             22222
                                                                 1111111111

Cf. A047966, A181819, A323023, A325247, A325260, A325262.
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).
Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

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

a(n) + A325262(n) = A000041(n).

A325610 Adjusted frequency depth of 2^n - 1.

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 1, 3, 3, 3, 3, 5, 1, 3, 3, 3, 1, 5, 1, 5, 5, 3, 3, 5, 3, 3, 3, 3, 3, 5, 1, 3, 3, 3, 3, 5, 3, 3, 3, 5, 3, 5, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 5, 1, 3, 5, 3, 3, 5, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 5, 3, 5, 3, 3, 3, 5, 3, 3, 3

Offset: 1

Gus Wiseman, May 12 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.

Cf. A001222, A001221, A056239, A071625, A112798, A323014, A325280.

Mersenne numbers: A046051, A046800, A059305, A325611, A325612, A325625.

fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#1]]&,ptn,Length[#1]>1&]]];
Table[fdadj[2^n-1],{n,100}]

A325283 Heinz numbers of integer partitions with maximum adjusted frequency depth for partitions of that sum.

Original entry on oeis.org

2, 4, 6, 12, 18, 20, 24, 28, 40, 48, 60, 84, 90, 120, 126, 132, 140, 150, 156, 168, 180, 198, 204, 220, 228, 234, 240, 252, 260, 264, 270, 276, 280

Offset: 1

Gus Wiseman, Apr 17 2019

The enumeration of these partitions by sum is given by A325254.
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 Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices and their omega-sequences (see A323023) begins:
  2:   {1}         (1)
  4:   {1,1}       (2,1)
  6:   {1,2}       (2,2,1)
  12:  {1,1,2}     (3,2,2,1)
  18:  {1,2,2}     (3,2,2,1)
  20:  {1,1,3}     (3,2,2,1)
  24:  {1,1,1,2}   (4,2,2,1)
  28:  {1,1,4}     (3,2,2,1)
  40:  {1,1,1,3}   (4,2,2,1)
  48:  {1,1,1,1,2} (5,2,2,1)
  60:  {1,1,2,3}   (4,3,2,2,1)
  84:  {1,1,2,4}   (4,3,2,2,1)
  90:  {1,2,2,3}   (4,3,2,2,1)
  120: {1,1,1,2,3} (5,3,2,2,1)
  126: {1,2,2,4}   (4,3,2,2,1)
  132: {1,1,2,5}   (4,3,2,2,1)
  140: {1,1,3,4}   (4,3,2,2,1)
  150: {1,2,3,3}   (4,3,2,2,1)
  156: {1,1,2,6}   (4,3,2,2,1)
  168: {1,1,1,2,4} (5,3,2,2,1)
  180: {1,1,2,2,3} (5,3,2,2,1)

Cf. A011784, A056239, A112798, A118914, A181819, A182857, A225486, A323014, A323023, A325254, A325258, A325277, A325278, A325282.

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

nn=30;
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
mfds=Table[Max@@fdadj/@IntegerPartitions[n],{n,nn}];
Select[Range[Prime[nn]],fdadj[primeMS[#]]==mfds[[Total[primeMS[#]]]]&]

A325336 Triangle read by rows where T(n,k) is the number of integer partitions of n with adjusted frequency depth k whose parts cover an initial interval of positive integers.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 0, 3, 1, 0, 0, 0, 0, 1, 0, 3, 2, 0, 0, 0, 0, 0, 1, 1, 3, 3, 0, 0, 0, 0, 0, 0, 1, 1, 5, 3, 0, 0, 0, 0, 0, 0, 0, 1, 0, 8, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 6, 6, 0, 0, 0

Offset: 0

Gus Wiseman, May 01 2019

The adjusted frequency depth of an integer partition (A325280) 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).

			Triangle begins:
  1
  0  1
  0  0  1
  0  0  1  1
  0  0  1  0  1
  0  0  1  0  2  0
  0  0  1  2  1  0  0
  0  0  1  0  3  1  0  0
  0  0  1  0  3  2  0  0  0
  0  0  1  1  3  3  0  0  0  0
  0  0  1  1  5  3  0  0  0  0  0
  0  0  1  0  8  3  0  0  0  0  0  0
  0  0  1  2  6  6  0  0  0  0  0  0  0
  0  0  1  0 13  4  0  0  0  0  0  0  0  0
  0  0  1  0 12  8  1  0  0  0  0  0  0  0  0
  0  0  1  2 14  7  3  0  0  0  0  0  0  0  0  0
  0  0  1  0 17 11  3  0  0  0  0  0  0  0  0  0  0
  0  0  1  0 22  7  8  0  0  0  0  0  0  0  0  0  0  0
  0  0  1  2 17 16 10  0  0  0  0  0  0  0  0  0  0  0  0
  0  0  1  0 28 10 15  0  0  0  0  0  0  0  0  0  0  0  0  0
  0  0  1  1 29 13 20  0  0  0  0  0  0  0  0  0  0  0  0  0  0
Row 15 counts the following partitions:
  111111111111111  54321       433221          333321        4322211
                   2222211111  443211          3332211       4332111
                               3322221         33222111      43221111
                               22222221        322221111
                               32222211        332211111
                               33321111        432111111
                               222222111       321111111111
                               3222111111
                               3321111111
                               22221111111
                               32211111111
                               222111111111
                               2211111111111
                               21111111111111

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A000009.
Column k = 3 is A325334.
Column k = 4 is A325335.
Cf. A181819, A182850, A317246, A320348, A323014, A325280, A325372.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#1]]&,ptn,Length[#1]>1&]]];
Table[Length[Select[IntegerPartitions[n],normQ[#]&&fdadj[#]==k&]],{n,0,30},{k,0,n}]

depth(p)={if(!#p, 0, my(r=1); 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)); listsort(L); p=L; r++); r)}
isok(p)={if(#p, for(i=1, #p, if(p[i]-1 > if(i>1, p[i-1], 0), return(0)))); 1}
row(n)={my(v=vector(1+n)); forpart(p=n, if(isok(p), v[1+depth(Vec(p))]++)); v}
{ for(n=0, 10, print(row(n))) } \\ Andrew Howroyd, Jan 18 2023

A325414 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with omega-sequence summing to n.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 1, 0, 1, 1, 2, 0, 3, 1, 1, 1, 0, 1, 0, 0, 0, 3, 0, 1, 4, 2, 2, 1, 1, 0, 1, 0, 1, 0, 4, 0, 3, 3, 2, 2, 2, 3, 1, 0, 1, 0, 0, 1, 4, 0, 3, 3, 3, 4, 1, 6, 3, 1, 0, 1

Offset: 0

Gus Wiseman, Apr 24 2019

The omega-sequence of an integer partition is the sequence of lengths of the multisets obtained by repeatedly taking the multiset of multiplicities until a singleton is reached. For example, the partition (32211) has chain of multisets of multiplicities {1,1,2,2,3} -> {1,2,2} -> {1,2} -> {1,1} -> {2}, so its omega-sequence is (5,3,2,2,1) with sum 13, so (32211) is counted under T(9,13).

			Triangle begins:
  1
  0 1
  0 1 0 1
  0 1 0 0 1 1
  0 1 0 1 0 2 0 0 1
  0 1 0 0 0 2 1 0 2 1
  0 1 0 1 1 2 0 3 1 1 1
  0 1 0 0 0 3 0 1 4 2 2 1 1
  0 1 0 1 0 4 0 3 3 2 2 2 3 1
  0 1 0 0 1 4 0 3 3 3 4 1 6 3 1
  0 1 0 1 0 4 1 6 4 4 1 4 5 8 2 1
Row n = 9 counts the following partitions:
  9  333  54  432  441  3222    22221      411111  3321     32211     321111
          63  531  522  6111    33111              4221     42111
          72  621  711  222111  51111              4311     21111111
          81                    111111111          5211
                                                   2211111
                                                   3111111

Row sums are A000041.
Row lengths are A325413(n) + 1 (because k starts at 0).
Number of nonzero terms in row n is A325415(n).
Cf. A181819, A225486, A323014, A323023, A325248, A325249, A325277, A325412, A325415, A325416.
Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (frequency depth), A325414 (omega-sequence sum).

omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];
Table[Length[Select[IntegerPartitions[n],Total[omseq[#]]==k&]],{n,0,10},{k,0,Max[Total/@omseq/@IntegerPartitions[n]]}]

cp's OEIS Frontend

A325254 Number of integer partitions of n with the maximum adjusted frequency depth for partitions of n.

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A377285 Position of first 0 in the n-th differences of the strict partition numbers A000009, or 0 if 0 does not appear.

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A353389 Create the sequence of all positive integers > 1 that are prime or whose prime shadow (A181819) is a divisor that is already in the sequence. Then remove all the primes.

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A182863 Members m of A025487 such that, if k appears in m's prime signature, k-1 appears at least as often as k (for any integer k > 1).

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A325246 Number of integer partitions of n with adjusted frequency depth equal to their length.

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A325250 Number of integer partitions of n whose omega-sequence is strict (no repeated parts).

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Formula

A325610 Adjusted frequency depth of 2^n - 1.

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A325283 Heinz numbers of integer partitions with maximum adjusted frequency depth for partitions of that sum.

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A325336 Triangle read by rows where T(n,k) is the number of integer partitions of n with adjusted frequency depth k whose parts cover an initial interval of positive integers.

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