This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A179009 #64 Jun 12 2025 10:24:25
%S A179009 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,
%T A179009 11,15,7,13,9,9,9,10,12,16,22,11,20,15,17,14,15,16,18,24,30,18,30,26,
%U A179009 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
%N A179009 Number of maximally refined partitions of n into distinct parts.
%C A179009 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.
%C A179009 The value of a(0) is taken to be 1 as is often done when considering partitions (also, the empty partition cannot be refined).
%C A179009 This sequence was suggested by _Moshe Shmuel Newman_.
%C A179009 From _Gus Wiseman_, Jun 07 2025: (Start)
%C A179009 Given any strict partition, the following are equivalent:
%C A179009 1) The parts are maximally refined.
%C A179009 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.
%C A179009 3) No part is a sum of distinct non-parts.
%C A179009 (End)
%H A179009 Massimo Lauria, <a href="/A179009/b179009.txt">Table of n, a(n) for n = 0..1500</a> (first 1000 terms by Fausto A. C. Cariboni)
%H A179009 Riccardo Aragona, Lorenzo Campioni, Roberto Civino, and Massimo Lauria, <a href="https://arxiv.org/abs/2111.11084">On the maximal part in unrefinable partitions of triangular numbers</a>, arXiv:2111.11084 [math.CO], 2021.
%H A179009 Riccardo Aragona, Roberto Civino, and Norberto Gavioli, <a href="https://arxiv.org/abs/2301.06347">A modular idealizer chain and unrefinability of partitions with repeated parts</a>, arXiv:2301.06347 [math.RA], 2023.
%H A179009 Riccardo Aragona, Roberto Civino, Norberto Gavioli and Carlo Maria Scoppola, <a href="https://arxiv.org/abs/2107.04666">Unrefinable partitions into distinct parts in a normalizer chain</a>, arXiv:2107.04666 [math.CO], 2021.
%H A179009 Riccardo Aragona, Lorenzo Campioni, Roberto Civino and Massimo Lauria, <a href="https://arxiv.org/abs/2112.15096">Verification and generation of unrefinable partitions</a>, arXiv:2112.15096 [math.CO], 2021.
%H A179009 Joerg Arndt, <a href="http://www.jjj.de/fxt/demo/seq/#A179009">C++ program</a> to compute such partitions.
%e A179009 a(11)=2 because there are two partitions of 11 which are maximally refined, namely 6+4+1 and 5+3+2+1.
%e A179009 From _Joerg Arndt_, Apr 23 2023: (Start)
%e A179009 The 15 maximally refined partitions of 35 are:
%e A179009 1: [ 1 2 3 4 5 6 14 ]
%e A179009 2: [ 1 2 3 4 5 7 13 ]
%e A179009 3: [ 1 2 3 4 5 8 12 ]
%e A179009 4: [ 1 2 3 4 5 9 11 ]
%e A179009 5: [ 1 2 3 4 6 7 12 ]
%e A179009 6: [ 1 2 3 4 6 8 11 ]
%e A179009 7: [ 1 2 3 4 6 9 10 ]
%e A179009 8: [ 1 2 3 4 7 8 10 ]
%e A179009 9: [ 1 2 3 5 6 7 11 ]
%e A179009 10: [ 1 2 3 5 6 8 10 ]
%e A179009 11: [ 1 2 3 5 7 8 9 ]
%e A179009 12: [ 1 2 4 5 6 7 10 ]
%e A179009 13: [ 1 2 4 5 6 8 9 ]
%e A179009 14: [ 1 3 4 5 6 7 9 ]
%e A179009 15: [ 2 3 4 5 6 7 8 ]
%e A179009 (End)
%t A179009 nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
%t A179009 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Intersection[#,Total/@nonsets[#]]=={}&]],{n,0,15}] (* _Gus Wiseman_, Jun 09 2025 *)
%Y A179009 For subsets instead of partitions we have A326080, complement A384350.
%Y A179009 These partitions are ranked by A383707, apparently positions of 1 in A383706.
%Y A179009 The strict complement is A384318 (strict partitions that can be refined).
%Y A179009 This is the strict version of A384392, ranks A384320, complement apparently A384321.
%Y A179009 Cf. A048767, A098859, A179822, A239455, A317142, A326083, A351293, A357982, A381454, A382525, A383708, A383710, A384317.
%K A179009 nonn
%O A179009 0,6
%A A179009 _David S. Newman_, Jan 03 2011
%E A179009 More terms from _Joerg Arndt_, Jan 04 2011