A325285 -id:A325285

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A325262

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

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).

LINKS

Table of n, a(n) for n=0..47.

EXAMPLE

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)

MATHEMATICA

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}]

CROSSREFS

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).

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 23 2019

STATUS

approved

A325411

Numbers whose omega-sequence has repeated parts.

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..65.

EXAMPLE

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

MATHEMATICA

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

Select[Range[100], !UnsameQ@@omseq[#]&]

CROSSREFS

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).