A384392 -id:A384392 - cp's OEIS frontend

A179009 Number of maximally refined partitions of n into distinct parts.

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 2, 3, 5, 1, 3, 2, 3, 5, 7, 2, 5, 3, 4, 6, 7, 11, 3, 8, 5, 6, 6, 8, 11, 15, 7, 13, 9, 9, 9, 10, 12, 16, 22, 11, 20, 15, 17, 14, 15, 16, 18, 24, 30, 18, 30, 26, 28, 22, 27, 21, 25, 27, 33, 42, 36, 45, 43, 46, 38, 44, 33, 43, 36, 44, 47, 60, 46, 66, 64, 70, 63, 72, 61, 69, 60, 63, 58, 69, 80

Offset: 0

David S. Newman, Jan 03 2011

nonn

Let a_1,a_2,...,a_k be a partition of n into distinct parts. We say that this partition can be refined if one of the summands, say a_i can be replaced with two numbers whose sum is a_i  and the resulting partition is a partition into distinct parts.  For example, the partition 5+2 can be refined because 5 can be replaced by 4+1 to give 4+2+1.  If a partition into distinct parts cannot be refined we say that it is maximally refined.
The value of a(0) is taken to be 1 as is often done when considering partitions (also, the empty partition cannot be refined).
This sequence was suggested by Moshe Shmuel Newman.
From Gus Wiseman, Jun 07 2025: (Start)
Given any strict partition, the following are equivalent:
1) The parts are maximally refined.
2) Every strict partition of a part contains a part. In other words, if y is the set of parts and z is any strict partition of any element of y, then z must contain at least one element from y.
3) No part is a sum of distinct non-parts.
(End)

			a(11)=2 because there are two partitions of 11 which are maximally refined, namely 6+4+1 and 5+3+2+1.
From _Joerg Arndt_, Apr 23 2023: (Start)
The 15 maximally refined partitions of 35 are:
   1:    [ 1 2 3 4 5 6 14 ]
   2:    [ 1 2 3 4 5 7 13 ]
   3:    [ 1 2 3 4 5 8 12 ]
   4:    [ 1 2 3 4 5 9 11 ]
   5:    [ 1 2 3 4 6 7 12 ]
   6:    [ 1 2 3 4 6 8 11 ]
   7:    [ 1 2 3 4 6 9 10 ]
   8:    [ 1 2 3 4 7 8 10 ]
   9:    [ 1 2 3 5 6 7 11 ]
  10:    [ 1 2 3 5 6 8 10 ]
  11:    [ 1 2 3 5 7 8 9 ]
  12:    [ 1 2 4 5 6 7 10 ]
  13:    [ 1 2 4 5 6 8 9 ]
  14:    [ 1 3 4 5 6 7 9 ]
  15:    [ 2 3 4 5 6 7 8 ]
(End)

Massimo Lauria, Table of n, a(n) for n = 0..1500 (first 1000 terms by Fausto A. C. Cariboni)
Riccardo Aragona, Lorenzo Campioni, Roberto Civino, and Massimo Lauria, On the maximal part in unrefinable partitions of triangular numbers, arXiv:2111.11084 [math.CO], 2021.
Riccardo Aragona, Roberto Civino, and Norberto Gavioli, A modular idealizer chain and unrefinability of partitions with repeated parts, arXiv:2301.06347 [math.RA], 2023.
Riccardo Aragona, Roberto Civino, Norberto Gavioli and Carlo Maria Scoppola, Unrefinable partitions into distinct parts in a normalizer chain, arXiv:2107.04666 [math.CO], 2021.
Riccardo Aragona, Lorenzo Campioni, Roberto Civino and Massimo Lauria, Verification and generation of unrefinable partitions, arXiv:2112.15096 [math.CO], 2021.
Joerg Arndt, C++ program to compute such partitions.

For subsets instead of partitions we have A326080, complement A384350.
These partitions are ranked by A383707, apparently positions of 1 in A383706.
The strict complement is A384318 (strict partitions that can be refined).
This is the strict version of A384392, ranks A384320, complement apparently A384321.
Cf. A048767, A098859, A179822, A239455, A317142, A326083, A351293, A357982, A381454, A382525, A383708, A383710, A384317.

nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Intersection[#,Total/@nonsets[#]]=={}&]],{n,0,15}] (* Gus Wiseman, Jun 09 2025 *)

More terms from Joerg Arndt, Jan 04 2011

A384318 Number of strict integer partitions of n that are not maximally refined.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 4, 4, 5, 9, 10, 13, 15, 17, 26, 29, 36, 43, 49, 57, 74, 84, 101, 118, 136, 158, 181, 219, 248, 291

Offset: 0

Gus Wiseman, May 28 2025

This is the number of strict integer partitions of n containing at least one sum of distinct non-parts.

Conjecture: Also the number of strict integer partitions of n such that it is possible in more than one way to choose a disjoint family of strict integer partitions, one of each part.

			For y = (5,4,2) we have 4 = 3+1 so y is counted under a(11).
On the other hand, no part of z = (6,4,1) is a subset-sum of the non-parts {2,3,5}, so z is not counted under a(11).
The a(3) = 1 through a(11) = 10 strict partitions:
  (3)  (4)  (5)  (6)    (7)    (8)    (9)    (10)     (11)
                 (4,2)  (4,3)  (5,3)  (5,4)  (6,4)    (6,5)
                 (5,1)  (5,2)  (6,2)  (6,3)  (7,3)    (7,4)
                        (6,1)  (7,1)  (7,2)  (8,2)    (8,3)
                                      (8,1)  (9,1)    (9,2)
                                             (5,3,2)  (10,1)
                                             (5,4,1)  (5,4,2)
                                             (6,3,1)  (6,3,2)
                                             (7,2,1)  (7,3,1)
                                                      (8,2,1)

The strict complement is A179009, ranks A383707.
The non-strict version for at least one choice is A383708, for none A383710.
The non-strict version is A384317, ranks A384321, complement A384392, ranks A384320.
These partitions are ranked by A384322.
For subsets instead of partitions we have A384350, complement A326080.
Cf. A357982, A383706 (disjoint), A384319, A384323 (non-strict).
Cf. A048767, A098859, A179822, A239455, A279375, A317142, A351293, A382525, A383533, A383711, A384391.

nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Intersection[#,Total/@nonsets[#]]!={}&]],{n,0,30}]

a(n) = A000009(n) - A179009(n).

A384320 Heinz numbers of integer partitions whose distinct parts are maximally refined.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 42, 45, 48, 50, 54, 56, 60, 64, 66, 70, 72, 75, 78, 80, 81, 84, 90, 96, 98, 100, 105, 108, 110, 112, 120, 126, 128, 132, 135, 140, 144, 150, 156, 160, 162, 168, 180, 182, 192, 196

Offset: 1

Gus Wiseman, Jun 01 2025

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
Given a partition, the following are equivalent:
1) The distinct parts are maximally refined.
2) Every strict partition of a part contains a part. In other words, if y is the set of parts and z is any strict partition of any element of y, then z must contain at least one element from y.
3) No part is a sum of distinct non-parts.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   27: {2,2,2}
   28: {1,1,4}
   30: {1,2,3}
   32: {1,1,1,1,1}

The squarefree case is A383707, counted by A179009.
The complement appears to be A384321, strict case A384322, counted by A384318.
Partitions of this type are counted by A384392.
A048767 is the Look-and-Say transform, fixed points A048768.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A383706, A357982 (non-disjoint), A299200 (non-strict).
Cf. A130091, A279375, A279790, A317142, A326080, A351294, A351295, A381454, A382525, A384390.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Select[Range[20],With[{y=Union[prix[#]]},UnsameQ@@y&&Intersection[y,Total/@nonsets[y]]=={}]&]

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A179009 Number of maximally refined partitions of n into distinct parts.

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A384318 Number of strict integer partitions of n that are not maximally refined.

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A384320 Heinz numbers of integer partitions whose distinct parts are maximally refined.

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