A325250 -id:A325250 - cp's OEIS frontend

A325268 Triangle read by rows where T(n,k) is the number of integer partitions of n with omicron k.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 0, 1, 0, 1, 5, 0, 0, 1, 0, 1, 7, 2, 0, 0, 1, 0, 1, 12, 1, 0, 0, 0, 1, 0, 1, 17, 2, 1, 0, 0, 0, 1, 0, 1, 24, 4, 0, 0, 0, 0, 0, 1, 0, 1, 33, 5, 1, 1, 0, 0, 0, 0, 1, 0, 1, 44, 9, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 57, 14, 3, 0, 1

Offset: 0

Gus Wiseman, Apr 18 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. The omicron of the partition is 0 if the omega-sequence is empty, 1 if it is a singleton, and otherwise the second-to-last part. 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), and its omicron is 2.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  3  0  1
  0  1  5  0  0  1
  0  1  7  2  0  0  1
  0  1 12  1  0  0  0  1
  0  1 17  2  1  0  0  0  1
  0  1 24  4  0  0  0  0  0  1
  0  1 33  5  1  1  0  0  0  0  1
  0  1 44  9  1  0  0  0  0  0  0  1
  0  1 57 14  3  0  1  0  0  0  0  0  1
  0  1 76 20  3  0  0  0  0  0  0  0  0  1
Row n = 8 counts the following partitions.
  (8)  (44)       (431)  (2222)  (11111111)
       (53)       (521)
       (62)
       (71)
       (332)
       (422)
       (611)
       (3221)
       (3311)
       (4211)
       (5111)
       (22211)
       (32111)
       (41111)
       (221111)
       (311111)
       (2111111)

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

Row sums are A000041. Column k = 2 is A325267.
Cf. A181819, A181821, A304634, A304636, A323014, A323023, A325250, A325273, 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).
Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

Table[Length[Select[IntegerPartitions[n],Switch[#,{},0,{},1,,NestWhile[Sort[Length/@Split[#]]&,#,Length[#]>1&]//First]==k&]],{n,0,10},{k,0,n}]

omicron(p)={if(!#p, 0, my(r=1); while(#p > 1, my(L=List(), k=0); r=#p; for(i=1, #p, if(i==#p||p[i]<>p[i+1], listput(L,i-k); k=i)); listsort(L); p=L); r)}
row(n)={my(v=vector(1+n)); forpart(p=n, v[1 + omicron(Vec(p))]++); v}
{ for(n=0, 10, print(row(n))) } \\ Andrew Howroyd, Jan 18 2023

A325247 Numbers whose omega-sequence is strict (no repeated parts).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 100, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193

Offset: 1

Gus Wiseman, Apr 16 2019

nonn

First differs from A323306 in having 216.
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).
Also Heinz numbers of integer partitions of whose omega-sequence is strict (counted by A325250). 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 begins:
     1: {}
     2: {1}
     3: {2}
     4: {1,1}
     5: {3}
     7: {4}
     8: {1,1,1}
     9: {2,2}
    11: {5}
    13: {6}
    16: {1,1,1,1}
    17: {7}
    19: {8}
    23: {9}
    25: {3,3}
    27: {2,2,2}
    29: {10}
    31: {11}
    32: {1,1,1,1,1}
    36: {1,1,2,2}

Positions of squarefree numbers in A325248.
Cf. A056239, A112798, A118914, A181819, A323023, A325249, A325250, A325251, A325277.
Omega-sequence statistics: A001221 (second omega), A001222 (first omega), A071625 (third omega), A304465 (second-to-last omega), A182850 or A323014 (depth), A323022 (fourth omega), A325248 (Heinz number).

omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#1]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
Select[Range[100],UnsameQ@@omseq[#]&]

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}]

A325267 Number of integer partitions of n with omicron 2.

Original entry on oeis.org

0, 0, 1, 1, 3, 5, 7, 12, 17, 24, 33, 44, 57, 76, 100, 129, 168, 214, 282, 355, 462, 586, 755, 937, 1202, 1493, 1900, 2349, 2944, 3621, 4520, 5514, 6813, 8298, 10150, 12240, 14918, 17931, 21654, 25917, 31081, 37029, 44256, 52474, 62405, 73724, 87378, 102887

Offset: 0

Gus Wiseman, Apr 18 2019

nonn

The Heinz numbers of these partitions are given by A304634.

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. We define the omicron of an integer partition to be 0 if the partition is empty, 1 if it is a singleton, and otherwise the second-to-last part of its omega-sequence. 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), and its omicron is 2.

			The a(1) = 1 through a(8) = 17 partitions:
  (11)  (21)  (22)   (32)    (33)     (43)      (44)
              (31)   (41)    (42)     (52)      (53)
              (211)  (221)   (51)     (61)      (62)
                     (311)   (411)    (322)     (71)
                     (2111)  (2211)   (331)     (332)
                             (3111)   (511)     (422)
                             (21111)  (2221)    (611)
                                      (3211)    (3221)
                                      (4111)    (3311)
                                      (22111)   (4211)
                                      (31111)   (5111)
                                      (211111)  (22211)
                                                (32111)
                                                (41111)
                                                (221111)
                                                (311111)
                                                (2111111)

Cf. A056239, A112798, A181819, A304634, A304636, A323014, A323023, A325250, A325273, 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).
Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

Table[Length[Select[IntegerPartitions[n],Switch[#,{},0,{},1,,NestWhile[Sort[Length/@Split[#]]&,#,Length[#]>1&]//First]==2&]],{n,0,30}]

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 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), which has repeated parts, so (32211) is counted under a(9).

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}]

A325411 Numbers whose omega-sequence has repeated parts.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 102, 104, 105

Offset: 1

Gus Wiseman, Apr 24 2019

nonn

First differs from A323304 in lacking 216. First differs from A106543 in having 144.
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 whose omega-sequence has repeated parts. The enumeration of these partitions by sum is given by A325285.
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), which has repeated parts, so 180 is in the sequence.

			The sequence of terms together with their omega-sequences begins:
   6: 2 2 1       51: 2 2 1         86: 2 2 1        119: 2 2 1
  10: 2 2 1       52: 3 2 2 1       87: 2 2 1        120: 5 3 2 2 1
  12: 3 2 2 1     54: 4 2 2 1       88: 4 2 2 1      122: 2 2 1
  14: 2 2 1       55: 2 2 1         90: 4 3 2 2 1    123: 2 2 1
  15: 2 2 1       56: 4 2 2 1       91: 2 2 1        124: 3 2 2 1
  18: 3 2 2 1     57: 2 2 1         92: 3 2 2 1      126: 4 3 2 2 1
  20: 3 2 2 1     58: 2 2 1         93: 2 2 1        129: 2 2 1
  21: 2 2 1       60: 4 3 2 2 1     94: 2 2 1        130: 3 3 1
  22: 2 2 1       62: 2 2 1         95: 2 2 1        132: 4 3 2 2 1
  24: 4 2 2 1     63: 3 2 2 1       96: 6 2 2 1      133: 2 2 1
  26: 2 2 1       65: 2 2 1         98: 3 2 2 1      134: 2 2 1
  28: 3 2 2 1     66: 3 3 1         99: 3 2 2 1      135: 4 2 2 1
  30: 3 3 1       68: 3 2 2 1      102: 3 3 1        136: 4 2 2 1
  33: 2 2 1       69: 2 2 1        104: 4 2 2 1      138: 3 3 1
  34: 2 2 1       70: 3 3 1        105: 3 3 1        140: 4 3 2 2 1
  35: 2 2 1       72: 5 2 2 1      106: 2 2 1        141: 2 2 1
  38: 2 2 1       74: 2 2 1        108: 5 2 2 1      142: 2 2 1
  39: 2 2 1       75: 3 2 2 1      110: 3 3 1        143: 2 2 1
  40: 4 2 2 1     76: 3 2 2 1      111: 2 2 1        144: 6 2 2 1
  42: 3 3 1       77: 2 2 1        112: 5 2 2 1      145: 2 2 1
  44: 3 2 2 1     78: 3 3 1        114: 3 3 1        146: 2 2 1
  45: 3 2 2 1     80: 5 2 2 1      115: 2 2 1        147: 3 2 2 1
  46: 2 2 1       82: 2 2 1        116: 3 2 2 1      148: 3 2 2 1
  48: 5 2 2 1     84: 4 3 2 2 1    117: 3 2 2 1      150: 4 3 2 2 1
  50: 3 2 2 1     85: 2 2 1        118: 2 2 1        152: 4 2 2 1

Positions of nonsquarefree numbers in A325248.
Cf. A056239, A112798, A118914, A181819, A323023, A325247, A325249, A325250, A325251, A325277, A325285.
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&]];
Select[Range[100],!UnsameQ@@omseq[#]&]

cp's OEIS Frontend

A325268 Triangle read by rows where T(n,k) is the number of integer partitions of n with omicron k.

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A325247 Numbers 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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Formula

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

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A325267 Number of integer partitions of n with omicron 2.

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

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A325411 Numbers whose omega-sequence has repeated parts.

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