A358335 -id:A358335 - cp's OEIS frontend

A141199 Number of hierarchical ordered partitions of partitions.

1, 1, 3, 7, 17, 38, 87, 191, 421, 911, 1963, 4186, 8885, 18724, 39284, 82005, 170521, 353214, 729290, 1501184, 3081869, 6311404, 12896983, 26301515, 53541702, 108815626, 220824295, 447524559, 905850001, 1831526719

Offset: 0

Thomas Wieder, Jun 13 2008, Jun 29 2008, Jul 28 2008

nonn

Consider the "ordered partitions of partitions" as described in A055887. They are produced by introducing separators (a term used by J. Riordan) between the parts of a partition. If a partition has P parts, then it is possible to introduce 1, 2, ... P-1 separators. Let "|" denote such a separator. We just append 1,2,...,P-1 separators to each integer partition of n and subsequently form all permutation of the resulting list (which is composed of parts and separators).
There are some rules: If we do not append a separator, then we do not perform any permutation. Furthermore, we do not accept permutations which have a dangling separator in front of the integer parts or past them. E.g. the permutations [|,1,2,3] and [1,2,3,|] are forbidden. Furthermore, sequences of separators as "|,|" are forbidden.
Now we impose a further restriction on the permutations. Consider the elements between two separators. We call their number "occupation number". We just request that the occupation number of a ordered partition is monotonically decreasing (if we start from the left to the right of a permutation written in our notation). If we interpret a separator as a level, then we can speak of a hierarchy. E.g. we do not count [1,|,2,3,|,4] as a hierarchy, but we accept [1,2|,3,4] as a hierarchy. We thus speak of "hierarchically ordered partitions of partitions" for this sequence.
With the generating function f := z -> 1/(mul(1-z^i/mul(1-z^j,j=1..i), i=1..25)); we get the asymptotic expansion using the command equivalent (f(z),z,n);
The result is 3.788561346*exp(-n)^(-log(2)) + O(1/n*exp(-n)^(-log(2))). Let fas := n -> 3.788562346*exp(-n)^(-log(2)); then for n=60 we get fas(60)/A141199(60)= .4367915009e19/4344507472742893655 = 1.005387846.
In short, a(n) is the number of finite sequences of integer partitions with weakly decreasing lengths and total sum n. The case of twice-partitions is A358831. A version choosing compositions is A218482. The strictly decreasing case is A358836. For ordered set partitions we have A005651. For weakly decreasing bigomega see A358335. - Gus Wiseman, Dec 05 2022

			n=1:
[1]
-------------------------
n=2:
[1, 1],
[1, "|", 1],
[2]
-------------------------
n=3:
[1, 2],
[1, "|", 1, "|", 1],
[1, 1, 1],
[3],
[2, "|", 1],
[1, 1, "|", 1],
[1, "|", 2]
-------------------------
n=4:
[1, 1, 1, "|", 1],
[1, 1, "|", 1, 1],
[2, 2],
[1, 3],
[1, 1, 1, 1],
[1, 1, 2],
[4],
[1, "|", 1, "|", 1, "|", 1],
[1, 2, "|", 1],
[1, 1, "|", 2],
[1, 1, "|", 1, "|", 1],
[2, "|", 1, "|", 1],
[1, "|", 2, "|", 1],
[1, "|", 1, "|", 2],
[1, "|", 3],
[3, "|", 1],
[2, "|", 2].

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Thomas Wieder, The number of certain rankings and hierarchies formed from labeled or unlabeled elements and sets, Applied Mathematical Sciences, vol. 3, 2009, no. 55, 2707 - 2724. [_Thomas Wieder_, Nov 14 2009]

Cf. A055887, A083355, A140585.

Cf. A000041, A000219, A001970, A005651, A063834, A129519, A218482, A358335, A358831, A358836, A358908.

A Maple program to generate these "hierarchically ordered partitions of partitions" is available on request.
An asymptotic expansion can be found using the generating function given by Vladeta Jovovic. For that purpose we use the Maple program "equivalent" from Bruno Salvy (http://ago.inria.fr/libraries/libraries.html).

my(N=40, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k/prod(j=1, k, 1-x^j))) \\ Seiichi Manyama, Jan 18 2022

G.f.: 1/Product_{i>=1} (1-x^i/Product_{j=1..i} (1-x^j)). - Vladeta Jovovic, Jul 16 2008

More terms from Vladeta Jovovic, Jul 16 2008

a(0)=1 prepended by Seiichi Manyama, Jan 18 2022

A358901 Number of integer partitions of n whose parts have all different numbers of prime factors (A001222).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 5, 7, 9, 8, 9, 11, 11, 15, 16, 16, 18, 20, 22, 26, 28, 31, 32, 36, 40, 45, 46, 46, 50, 59, 64, 70, 75, 78, 83, 89, 94, 108, 106, 104, 120, 137, 142, 147, 150, 161, 174, 190, 200, 220, 226, 224, 248, 274, 274, 287, 301, 320, 340, 351, 361

Offset: 0

Gus Wiseman, Dec 07 2022

nonn

			The a(1) = 1 through a(11) = 7 partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)    (B)
            (21)  (31)  (41)  (42)  (43)   (62)   (54)   (82)   (74)
                              (51)  (61)   (71)   (63)   (91)   (65)
                                    (421)  (431)  (81)   (451)  (83)
                                                  (621)  (631)  (92)
                                                                (A1)
                                                                (821)

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 101 terms from Lucas A. Brown)
Lucas A. Brown, Python program.

The weakly decreasing version is A358909 (complement A358910).
The version not counting multiplicity is A358903, weakly decreasing A358902.
For equal numbers of prime factors we have A319169, compositions A358911.
A001222 counts prime factors, distinct A001221.
A063834 counts twice-partitions.
A358836 counts multiset partitions with all distinct block sizes.
Cf. A056239, A129519, A141199, A218482, A300335, A319071, A320324, A358335, A358831, A358908.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@PrimeOmega/@#&]],{n,0,60}]

a(61) and beyond from Lucas A. Brown, Dec 14 2022

A358911 Number of integer compositions of n whose parts all have the same number of prime factors, counted with multiplicity.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 7, 9, 12, 20, 21, 39, 49, 79, 109, 161, 236, 345, 512, 752, 1092, 1628, 2376, 3537, 5171, 7650, 11266, 16634, 24537, 36173, 53377, 78791, 116224, 171598, 253109, 373715, 551434, 814066, 1201466, 1773425, 2617744, 3864050, 5703840, 8419699

Offset: 0

Gus Wiseman, Dec 11 2022

nonn

			The a(1) = 1 through a(8) = 9 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (23)     (33)      (25)       (35)
                    (1111)  (32)     (222)     (52)       (44)
                            (11111)  (111111)  (223)      (53)
                                               (232)      (233)
                                               (322)      (323)
                                               (1111111)  (332)
                                                          (2222)
                                                          (11111111)

Alois P. Heinz, Table of n, a(n) for n = 0..4000 (first 101 terms from Lucas A. Brown)
Lucas A. Brown, Python program.

The case of partitions is A319169, ranked by A320324.
The weakly decreasing version is A358335, strictly A358901.
For sequences of partitions see A358905.
A001222 counts prime factors, distinct A001221.
A011782 counts compositions.
A358902 = compositions with weakly decreasing A001221, strictly A358903.
A358909 = partitions with weakly decreasing A001222, complement A358910.
Cf. A056239, A063834, A064573, A218482, A279787, A300335, A319066, A319071, A358831, A358908.

b:= proc(n, i) option remember; uses numtheory; `if`(n=0, 1, add(
     (t-> `if`(i<0 or i=t, b(n-j, t), 0))(bigomega(j)), j=1..n))
    end:
a:= n-> b(n, -1):
seq(a(n), n=0..44);  # Alois P. Heinz, Feb 12 2024

Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n],SameQ@@PrimeOmega/@#&]],{n,0,10}]

a(21) and beyond from Lucas A. Brown, Dec 15 2022

A358903 Number of integer partitions of n whose parts have all different numbers of distinct prime factors (A001221).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 5, 7, 8, 7, 9, 10, 10, 10, 9, 11, 15, 14, 13, 15, 14, 14, 17, 16, 17, 17, 16, 16, 17, 17, 21, 26, 24, 23, 25, 27, 29, 32, 31, 29, 36, 36, 35, 37, 37, 42, 49, 45, 44, 50, 49, 50, 58, 55, 55, 58, 56, 58, 66, 62, 65, 75

Offset: 0

Gus Wiseman, Dec 07 2022

nonn

			The a(15) = 8 partitions are: (15), (14,1), (12,3), (12,2,1), (10,5), (10,4,1), (6,9), (8,6,1).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 101 terms from Lucas A. Brown)
Lucas A. Brown, Python program.

Counting prime factors with multiplicity gives A358901.
The weakly decreasing version is A358902, with multiplicity A358335.
A001222 counts prime factors, distinct A001221.
A116608 counts partitions by sum and number of distinct parts.
A358836 counts multiset partitions with all distinct block sizes.
Cf. A046660, A071625, A129519, A141199, A319169, A358831, A358909, A358911.

p:= proc(n) option remember; nops(ifactors(n)[2]) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
      add((t-> `if`(t b(n$2):
seq(a(n), n=0..68);  # Alois P. Heinz, Feb 14 2024

Table[Length[Select[IntegerPartitions[n],UnsameQ@@PrimeNu/@#&]],{n,0,30}]

a(56) and beyond from Lucas A. Brown, Dec 14 2022

A358909 Number of integer partitions of n whose parts have weakly decreasing numbers of prime factors (A001222).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, 53, 73, 93, 124, 157, 206, 256, 329, 406, 514, 628, 784, 949, 1174, 1411, 1725, 2061, 2500, 2966, 3570, 4217, 5039, 5919, 7027, 8219, 9706, 11301, 13268, 15394, 17995, 20792, 24195, 27863, 32288, 37061, 42779, 48950, 56306

Offset: 0

Gus Wiseman, Dec 09 2022

nonn

First differs from A000041 at a(9) = 29, A000041(9) = 30, the difference coming from the partition (5,4).

For sequences of partitions see A141199, compositions A218482.
The case of equality is A319169, for compositions A358911.
The case of compositions is A358335, strictly decreasing A358901.
The complement is counted by A358910.
A001222 counts prime factors, distinct A001221.
A011782 counts compositions.
A063834 counts twice-partitions.
Cf. A056239, A300335, A320324, A358831, A358903, A358908.

Table[Length[Select[IntegerPartitions[n],GreaterEqual@@PrimeOmega/@#&]],{n,0,30}]

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A141199 Number of hierarchical ordered partitions of partitions.

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A358901 Number of integer partitions of n whose parts have all different numbers of prime factors (A001222).

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A358911 Number of integer compositions of n whose parts all have the same number of prime factors, counted with multiplicity.

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A358903 Number of integer partitions of n whose parts have all different numbers of distinct prime factors (A001221).

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A358909 Number of integer partitions of n whose parts have weakly decreasing numbers of prime factors (A001222).

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