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A384176 Number of subsets of {1..n} without all distinct lengths of maximal runs (increasing by 1).

%I A384176 #7 Jun 17 2025 09:18:05
%S A384176 0,0,0,1,3,8,20,51,121,276,612,1335,2881,6144,12950,27029,55977,
%T A384176 115222,236058,481683,979443
%N A384176 Number of subsets of {1..n} without all distinct lengths of maximal runs (increasing by 1).
%e A384176 The subset {1,3,4,8,9} has maximal runs ((1),(3,4),(8,9)), with lengths (1,2,2), so is counted under a(10).
%e A384176 The a(0) = 0 through a(6) = 20 subsets:
%e A384176   .  .  .  {1,3}  {1,3}  {1,3}      {1,3}
%e A384176                   {1,4}  {1,4}      {1,4}
%e A384176                   {2,4}  {1,5}      {1,5}
%e A384176                          {2,4}      {1,6}
%e A384176                          {2,5}      {2,4}
%e A384176                          {3,5}      {2,5}
%e A384176                          {1,3,5}    {2,6}
%e A384176                          {1,2,4,5}  {3,5}
%e A384176                                     {3,6}
%e A384176                                     {4,6}
%e A384176                                     {1,3,5}
%e A384176                                     {1,3,6}
%e A384176                                     {1,4,6}
%e A384176                                     {2,4,6}
%e A384176                                     {1,2,4,5}
%e A384176                                     {1,2,4,6}
%e A384176                                     {1,2,5,6}
%e A384176                                     {1,3,4,6}
%e A384176                                     {1,3,5,6}
%e A384176                                     {2,3,5,6}
%t A384176 Table[Length[Select[Subsets[Range[n]],!UnsameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}]
%Y A384176 For equal instead of distinct lengths the complement is A243815.
%Y A384176 These subsets are ranked by the non-members of A328592.
%Y A384176 The complement is counted by A384175.
%Y A384176 For strict partitions instead of subsets see A384178, A384884, A384886, A384880.
%Y A384176 For permutations instead of subsets see A384891, A384892, A010027.
%Y A384176 A034839 counts subsets by number of maximal runs, for strict partitions A116674.
%Y A384176 A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
%Y A384176 A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
%Y A384176 Cf. A000009, A044813, A242882, A325325, A329739, A351202, A383013, A384177, A384889, A384890.
%K A384176 nonn,more
%O A384176 0,5
%A A384176 _Gus Wiseman_, Jun 16 2025