A325280 -id:A325280 - cp's OEIS frontend

A353845 Number of integer partitions of n such that if you repeatedly take the multiset of run-sums (or condensation), you eventually reach an empty set or singleton.

1, 1, 2, 2, 4, 2, 5, 2, 8, 3, 5, 2, 15, 2, 5, 4, 18, 2, 13, 2, 14, 4, 5, 2, 62, 3, 5, 5, 14, 2, 18, 2, 48, 4, 5, 4, 71, 2, 5, 4, 54, 2, 18, 2, 14, 10, 5, 2, 374, 3, 9, 4, 14, 2, 37, 4, 54, 4, 5, 2, 131

Offset: 0

Gus Wiseman, May 26 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).

			The a(1) = 1 through a(8) = 8 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                    (211)            (222)                (422)
                    (1111)           (3111)               (2222)
                                     (111111)             (4211)
                                                          (41111)
                                                          (221111)
                                                          (11111111)
For example, the partition (3,2,2,2,1,1,1) has trajectory: (1,1,1,2,2,2,3) -> (3,3,6) -> (6,6) -> (12), so is counted under a(12).

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Dominated by A018818 (partitions into divisors).
The version for compositions is A353858.
A275870 counts collapsible partitions, ranked by A300273.
A304442 counts partitions with all equal run-sums, ranked by A353833.
A325268 counts partitions by omicron, rank statistic A304465.
A353832 represents the operation of taking run-sums of a partition.
A353837 counts partitions with all distinct run-sums, ranked by A353838.
A353840-A353846 pertain to partition run-sum trajectory.
A353847-A353859 pertain to composition run-sum trajectory.
A353864 counts rucksack partitions, ranked by A353866.
Cf. A000041, A008284, A181819, A225485, A325239, A325277, A325280, A326370, A353834, A353839, A353865.

Table[Length[Select[IntegerPartitions[n], Length[NestWhile[Sort[Total/@Split[#]]&,#,!UnsameQ@@#&]]<=1&]],{n,0,30}]

A325260 Number of integer partitions of n whose omega-sequence covers an initial interval of positive integers.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 5, 8, 10, 12, 13, 18, 19, 24, 25, 31, 33, 40, 40, 49, 51, 59, 60, 71, 72, 83, 84, 96, 98, 111, 111, 126, 128, 142, 143, 160, 161, 178, 179, 197, 199, 218, 218, 239, 241, 261, 262, 285, 286, 309, 310, 334, 336, 361, 361, 388, 390, 416, 417, 446

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 A325251.

			The a(1) = 1 through a(9) = 12 partitions:
  (1)  (2)   (3)   (4)    (5)    (6)    (7)     (8)     (9)
       (11)  (21)  (22)   (32)   (33)   (43)    (44)    (54)
                   (31)   (41)   (42)   (52)    (53)    (63)
                   (211)  (221)  (51)   (61)    (62)    (72)
                          (311)  (411)  (322)   (71)    (81)
                                        (331)   (332)   (441)
                                        (511)   (422)   (522)
                                        (3211)  (611)   (711)
                                                (3221)  (3321)
                                                (4211)  (4221)
                                                        (4311)
                                                        (5211)

Cf. A055932, A181819, A182850, A225486, A323014, A323023, A325250, A325251, A325262, A325277.

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

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

a(n) + A325262(n) = A000041(n).
Conjectures from Chai Wah Wu, Jan 13 2021: (Start)
a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9) for n > 9.
G.f.: (-x^9 - x^8 - x^7 + x^6 - x^5 - x^2 - x - 1)/((x - 1)^3*(x + 1)^2*(x^2 + 1)*(x^2 + x + 1)). (End)

A325262 Number of integer partitions of n whose omega-sequence does not cover an initial interval of positive integers.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 6, 7, 12, 18, 29, 38, 58, 77, 110, 145, 198, 257, 345, 441, 576, 733, 942, 1184, 1503, 1875, 2352, 2914, 3620, 4454, 5493, 6716, 8221, 10001, 12167, 14723, 17816, 21459, 25836, 30988, 37139, 44365, 52956, 63022, 74934, 88873, 105296, 124469

Offset: 0

Gus Wiseman, Apr 23 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 a(3) = 1 through a(9) = 18 partitions:
  (111)  (1111)  (2111)   (222)     (421)      (431)       (333)
                 (11111)  (321)     (2221)     (521)       (432)
                          (2211)    (4111)     (2222)      (531)
                          (3111)    (22111)    (3311)      (621)
                          (21111)   (31111)    (5111)      (3222)
                          (111111)  (211111)   (22211)     (6111)
                                    (1111111)  (32111)     (22221)
                                               (41111)     (32211)
                                               (221111)    (33111)
                                               (311111)    (42111)
                                               (2111111)   (51111)
                                               (11111111)  (222111)
                                                           (321111)
                                                           (411111)
                                                           (2211111)
                                                           (3111111)
                                                           (21111111)
                                                           (111111111)

Cf. A055932, A181819, A182850, A225486, A323014, A323023, A325250, A325251, A325261, A325277, A325285.

Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (frequency depth), A325249 (sum).

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

A325374 Numbers with adjusted frequency depth 3 whose prime indices cover an initial interval of positive integers.

Original entry on oeis.org

6, 30, 36, 210, 216, 900, 1296, 2310, 7776, 27000, 30030, 44100, 46656, 279936, 510510, 810000, 1679616, 5336100, 9261000, 9699690, 10077696, 24300000, 60466176, 223092870, 362797056, 729000000, 901800900, 1944810000, 2176782336, 6469693230, 12326391000

Offset: 1

Gus Wiseman, May 02 2019

nonn

The adjusted frequency depth (A323014) 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.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions with adjusted frequency depth 3 whose parts cover an initial interval of positive integers. The enumeration of these partitions by sum is given by A325334.
The terms are the primorial numbers (A002110) above 2 and all their powers. - Amiram Eldar, May 08 2019

			The sequence of terms together with their prime indices begins:
      6: {1,2}
     30: {1,2,3}
     36: {1,1,2,2}
    210: {1,2,3,4}
    216: {1,1,1,2,2,2}
    900: {1,1,2,2,3,3}
   1296: {1,1,1,1,2,2,2,2}
   2310: {1,2,3,4,5}
   7776: {1,1,1,1,1,2,2,2,2,2}
  27000: {1,1,1,2,2,2,3,3,3}
  30030: {1,2,3,4,5,6}
  44100: {1,1,2,2,3,3,4,4}
  46656: {1,1,1,1,1,1,2,2,2,2,2,2}

Amiram Eldar, Table of n, a(n) for n = 1..5292

Cf. A002110, A055932, A056239, A112798, A130091, A181819, A320348, A323014, A325280, A325326, A325336, A325387.

normQ[n_Integer]:=Or[n==1,PrimePi/@First/@FactorInteger[n]==Range[PrimeNu[n]]];
fdadj[n_Integer]:=If[n==1,0,Length[NestWhileList[Times@@Prime/@Last/@FactorInteger[#1]&,n,!PrimeQ[#1]&]]];
Select[Range[10000],normQ[#]&&fdadj[#]==3&]

A325413 Largest sum of the omega-sequence of an integer partition of n.

Original entry on oeis.org

0, 1, 3, 5, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 0

Gus Wiseman, Apr 24 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) with sum 13.

Appears to contain all nonnegative integers except 2, 4, 6, 7, and 11.

			The partitions of 9 organized by sum of omega-sequence (first column) are:
   1: (9)
   4: (333)
   5: (81) (72) (63) (54)
   7: (621) (531) (432)
   8: (711) (522) (441)
   9: (6111) (3222) (222111)
  10: (51111) (33111) (22221) (111111111)
  11: (411111)
  12: (5211) (4311) (4221) (3321) (3111111) (2211111)
  13: (42111) (32211) (21111111)
  14: (321111)
The largest term in the first column is 14, so a(9) = 14.

Row lengths of A325414.
Cf. A181819, A225486, A323014, A323023, A325238, A325249, 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[Max[Total/@omseq/@IntegerPartitions[n]],{n,0,30}]

A325415 Number of distinct sums of omega-sequences of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 8, 10, 11, 13, 12, 15, 14, 16, 18, 18, 18, 21, 20, 23, 23, 24, 24, 27, 27, 28, 29, 30, 30, 34, 32, 34, 35, 36, 37, 39, 38, 40, 41, 43, 42, 45, 44, 46, 48, 48, 48, 51, 50, 53, 53, 54, 54, 57, 57, 58, 59, 60, 60, 64

Offset: 0

Gus Wiseman, Apr 24 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) with sum 13.

			The partitions of 9 organized by sum of omega sequence (first column) are:
   1: (9)
   4: (333)
   5: (81) (72) (63) (54)
   7: (621) (531) (432)
   8: (711) (522) (441)
   9: (6111) (3222) (222111)
  10: (51111) (33111) (22221) (111111111)
  11: (411111)
  12: (5211) (4311) (4221) (3321) (3111111) (2211111)
  13: (42111) (32211) (21111111)
  14: (321111)
There are a total of 11 distinct sums {1,4,5,7,8,9,10,11,12,13,14}, so a(9) = 11.

Number of nonzero terms in row n of A325414.
Cf. A181819, A225486, A323014, A323023, A325238, A325248, A325249, A325277, A325412, A325413, 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[Union[Total/@omseq/@IntegerPartitions[n]]],{n,0,30}]

A325416 Least k such that the omega-sequence of k sums to n, and 0 if none exists.

Original entry on oeis.org

1, 2, 0, 4, 8, 6, 32, 30, 12, 24, 48, 96, 60, 120, 240, 480, 960, 1920, 3840, 2520, 5040, 10080, 20160, 40320, 80640

Offset: 0

Gus Wiseman, Apr 25 2019

nonn

We define the omega-sequence of n (row n of A323023) to have length A323014(n) = adjusted frequency depth of n, and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, defined by red(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. For example, we have 180 -> 18 -> 6 -> 4 -> 3, so the omega-sequence of 180 is (5,3,2,2,1) with sum 13.

			The sequence of terms together with their omega-sequences (n = 2 term not shown) begins:
     1:
     2:  1
     4:  2 1
     8:  3 1
     6:  2 2 1
    32:  5 1
    30:  3 3 1
    12:  3 2 2 1
    24:  4 2 2 1
    48:  5 2 2 1
    96:  6 2 2 1
    60:  4 3 2 2 1
   120:  5 3 2 2 1
   240:  6 3 2 2 1
   480:  7 3 2 2 1
   960:  8 3 2 2 1
  1920:  9 3 2 2 1
  3840: 10 3 2 2 1
  2520:  7 4 3 2 2 1
  5040:  8 4 3 2 2 1

Cf. A056239, A181819, A181821, A304465, A307734, A323023, A325238, A325277, A325280, A325413, A325415.

Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (frequency depth), A325248 (Heinz number), A325249 (sum).

omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
da=Table[Total[omseq[n]],{n,10000}];
Table[If[!MemberQ[da,k],0,Position[da,k][[1,1]]],{k,0,Max@@da}]

A353698 Number of integer partitions of n whose product equals their length.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 2, 1, 2, 0, 2, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 3, 0, 3, 2, 2, 1, 5, 0, 1, 2, 5, 1, 4, 0, 3, 3, 2, 1, 4, 2, 3, 2, 2, 0, 5, 1, 4, 2, 2, 3, 6, 1, 2, 2, 5, 1, 4, 0, 4, 3, 3, 1, 6, 2, 3, 4, 4, 2, 4, 1, 4, 2, 3, 1, 8, 2, 4, 2, 4, 2, 5, 2, 4, 2

Offset: 0

Gus Wiseman, May 19 2022

nonn

			The a(n) partitions for selected n (A..H = 10..17):
n=9:    n=21:             n=27:                 n=33:
---------------------------------------------------------------------------
51111   B1111111111       E1111111111111        H1111111111111111
321111  72111111111111    921111111111111111    B211111111111111111111
        531111111111111   54111111111111111111  831111111111111111111111
        4221111111111111                        5511111111111111111111111
                                                333111111111111111111111111

Andrew Howroyd, Table of n, a(n) for n = 0..10000

The LHS (product of parts) is counted by A339095, rank statistic A003963.
The RHS (length) is counted by A008284, rank statistic A001222.
These partitions are ranked by A353699.
A266477 counts partitions by product of multiplicities, rank stat A005361.
A353504 counts partitions w/ product less than product of multiplicities.
A353505 counts partitions w/ product greater than product of multiplicities.
A353506 counts partitions w/ prod equal to prod of mults, ranked by A353503.
Cf. A000041, A002033, A098859, A114640, A181819, A225485, A266499, A319000, A325280, A353398, A353507.

Table[Length[Select[IntegerPartitions[n],Times@@#==Length[#]&]],{n,0,30}]

a(r,m=r,p=1,k=0) = {(p==k+r) + sum(m=2, min(m, (k+r)\p),  self()(r-m, min(m,r-m), p*m, k+1))} \\ Andrew Howroyd, Jan 02 2023

Terms a(61) and beyond from Andrew Howroyd, Jan 02 2023

A325335 Number of integer partitions of n with adjusted frequency depth 4 whose parts cover an initial interval of positive integers.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 1, 3, 3, 3, 5, 8, 6, 13, 12, 14, 17, 22, 17, 28, 29, 30, 38, 50, 46, 67, 64, 75, 81, 104, 99, 127, 128, 150, 155, 201, 189, 236, 244, 293, 302, 363, 372, 437, 457, 548, 547, 638, 671, 754, 809, 922, 947, 1074, 1144, 1290, 1342, 1515, 1574

Offset: 0

Gus Wiseman, May 01 2019

nonn

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).

The Heinz numbers of these partitions are given by A325387.

			The a(4) = 1 through a(10) = 5 partitions:
  (211)  (221)   (21111)  (2221)    (22211)    (22221)     (222211)
         (2111)           (22111)   (221111)   (2211111)   (322111)
                          (211111)  (2111111)  (21111111)  (2221111)
                                                           (22111111)
                                                           (211111111)

Column k = 4 of A325336.

Cf. A000009, A007862, A181819, A182850, A317081, A320348, A323014, A325280, A325326, A325334, A325387.

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[#]==4&]],{n,0,30}]

A325412 Number of distinct omega-sequences of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 5, 10, 9, 14, 15, 20, 21, 33, 30, 39, 45, 54, 54, 69, 68, 85, 90, 100, 104, 128, 127, 141, 153, 172, 175, 205, 203, 229, 240, 257, 274, 308, 309, 335, 356, 390, 395, 437, 444, 481, 506, 530, 549, 602, 609, 648, 672, 710, 727, 777, 798, 848, 871

Offset: 0

Gus Wiseman, Apr 24 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 a(1) = 1 through a(9) = 15 omega-sequences:
  (1)  (1)   (1)    (1)     (1)     (1)     (1)      (1)      (1)
       (21)  (31)   (21)    (51)    (21)    (71)     (21)     (31)
             (221)  (41)    (221)   (31)    (221)    (41)     (91)
                    (221)   (3221)  (61)    (331)    (81)     (221)
                    (3221)  (4221)  (221)   (3221)   (221)    (331)
                                    (331)   (4221)   (331)    (621)
                                    (421)   (5221)   (421)    (3221)
                                    (3221)  (6221)   (3221)   (4221)
                                    (4221)  (43221)  (4221)   (5221)
                                    (5221)           (5221)   (6221)
                                                     (6221)   (7221)
                                                     (7221)   (8221)
                                                     (43221)  (43221)
                                                     (53221)  (53221)
                                                              (63221)

Cf. A181819, A225486, A323014, A323023, A325238, A325248, A325277, A325413, A325415.

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[Union[omseq/@IntegerPartitions[n]]],{n,0,30}]

cp's OEIS Frontend

A353845 Number of integer partitions of n such that if you repeatedly take the multiset of run-sums (or condensation), you eventually reach an empty set or singleton.

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A325260 Number of integer partitions of n whose omega-sequence covers an initial interval of positive integers.

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A325262 Number of integer partitions of n whose omega-sequence does not cover an initial interval of positive integers.

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A325374 Numbers with adjusted frequency depth 3 whose prime indices cover an initial interval of positive integers.

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A325413 Largest sum of the omega-sequence of an integer partition of n.

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A325415 Number of distinct sums of omega-sequences of integer partitions of n.

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A325416 Least k such that the omega-sequence of k sums to n, and 0 if none exists.

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A353698 Number of integer partitions of n whose product equals their length.

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A325335 Number of integer partitions of n with adjusted frequency depth 4 whose parts cover an initial interval of positive integers.

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