A325262 -id:A325262 - cp's OEIS frontend

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

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).

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 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)

A325261 Numbers whose omega-sequence does not cover an initial interval of positive integers.

Original entry on oeis.org

8, 16, 24, 27, 30, 32, 36, 40, 42, 48, 54, 56, 64, 66, 70, 72, 78, 80, 81, 88, 96, 100, 102, 104, 105, 108, 110, 112, 114, 120, 125, 128, 130, 135, 136, 138, 144, 152, 154, 160, 162, 165, 168, 170, 174, 176, 180, 182, 184, 186, 189, 190, 192, 195, 196, 200

Offset: 1

Gus Wiseman, Apr 23 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).

			The sequence of terms together with their omega sequences begins:
    8: 3->1           108: 5->2->2->1        189: 4->2->2->1
   16: 4->1           110: 3->3->1           190: 3->3->1
   24: 4->2->2->1     112: 5->2->2->1        192: 7->2->2->1
   27: 3->1           114: 3->3->1           195: 3->3->1
   30: 3->3->1        120: 5->3->2->2->1     196: 4->2->1
   32: 5->1           125: 3->1              200: 5->2->2->1
   36: 4->2->1        128: 7->1              208: 5->2->2->1
   40: 4->2->2->1     130: 3->3->1           210: 4->4->1
   42: 3->3->1        135: 4->2->2->1        216: 6->2->1
   48: 5->2->2->1     136: 4->2->2->1        222: 3->3->1
   54: 4->2->2->1     138: 3->3->1           224: 6->2->2->1
   56: 4->2->2->1     144: 6->2->2->1        225: 4->2->1
   64: 6->1           152: 4->2->2->1        230: 3->3->1
   66: 3->3->1        154: 3->3->1           231: 3->3->1
   70: 3->3->1        160: 6->2->2->1        232: 4->2->2->1
   72: 5->2->2->1     162: 5->2->2->1        238: 3->3->1
   78: 3->3->1        165: 3->3->1           240: 6->3->2->2->1
   80: 5->2->2->1     168: 5->3->2->2->1     243: 5->1
   81: 4->1           170: 3->3->1           246: 3->3->1
   88: 4->2->2->1     174: 3->3->1           248: 4->2->2->1
   96: 6->2->2->1     176: 5->2->2->1        250: 4->2->2->1
  100: 4->2->1        180: 5->3->2->2->1     252: 5->3->2->2->1
  102: 3->3->1        182: 3->3->1           255: 3->3->1
  104: 4->2->2->1     184: 4->2->2->1        256: 8->1
  105: 3->3->1        186: 3->3->1           258: 3->3->1

Complement of A325251.
Cf. A000430, A055932, A056239, A112798, A181819, A323023, A325247, A325260, A325262, A325277.
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), A325249 (sum).

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
Select[Range[100],!normQ[omseq[#]]&]

A325285 Number of integer partitions of n whose omega-sequence has repeated parts.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 6, 13, 17, 26, 36, 54, 66, 98, 125, 164, 214, 285, 354, 468, 585, 745, 945, 1195, 1477, 1864, 2317, 2867, 3544, 4383, 5348, 6589, 8028, 9778, 11885, 14403, 17362, 20992, 25212, 30239, 36158, 43242, 51408, 61240, 72568, 85989, 101607, 120027

Offset: 0

Gus Wiseman, Apr 24 2019

nonn

The Heinz numbers of these partitions are given by A325411.

			The a(3) = 1 through a(8) = 17 partitions:
  (21)  (31)   (32)    (42)     (43)      (53)
        (211)  (41)    (51)     (52)      (62)
               (221)   (321)    (61)      (71)
               (311)   (411)    (322)     (332)
               (2111)  (3111)   (331)     (422)
                       (21111)  (421)     (431)
                                (511)     (521)
                                (2221)    (611)
                                (3211)    (3221)
                                (4111)    (4211)
                                (22111)   (5111)
                                (31111)   (22211)
                                (211111)  (32111)
                                          (41111)
                                          (221111)
                                          (311111)
                                          (2111111)

Cf. A047966, A181819, A323014, A323023, A325247, A325250, A325260, A325262, A325411.

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],!UnsameQ@@omseq[#]&]],{n,0,30}]

cp's OEIS Frontend

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

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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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A325261 Numbers whose omega-sequence does not cover an initial interval of positive integers.

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A325285 Number of integer partitions of n whose omega-sequence has repeated parts.

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