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A384175 Number of subsets of {1..n} with all distinct lengths of maximal runs (increasing by 1).

%I A384175 #12 Jun 18 2025 23:19:55
%S A384175 1,2,4,7,13,24,44,77,135,236,412,713,1215,2048,3434,5739,9559,15850,
%T A384175 26086,42605,69133,111634,179602,288069,460553,733370,1162356,1833371,
%U A384175 2878621,4501856,7016844,10905449,16904399,26132460,40279108,61885621,94766071,144637928
%N A384175 Number of subsets of {1..n} with all distinct lengths of maximal runs (increasing by 1).
%H A384175 Christian Sievers, <a href="/A384175/b384175.txt">Table of n, a(n) for n = 0..1000</a>
%e A384175 The subset {2,3,5,6,7,9} has maximal runs ((2,3),(5,6,7),(9)), with lengths (2,3,1), so is counted under a(9).
%e A384175 The a(0) = 1 through a(4) = 13 subsets:
%e A384175   {}  {}   {}     {}       {}
%e A384175       {1}  {1}    {1}      {1}
%e A384175            {2}    {2}      {2}
%e A384175            {1,2}  {3}      {3}
%e A384175                   {1,2}    {4}
%e A384175                   {2,3}    {1,2}
%e A384175                   {1,2,3}  {2,3}
%e A384175                            {3,4}
%e A384175                            {1,2,3}
%e A384175                            {1,2,4}
%e A384175                            {1,3,4}
%e A384175                            {2,3,4}
%e A384175                            {1,2,3,4}
%t A384175 Table[Length[Select[Subsets[Range[n]],UnsameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}]
%o A384175 (PARI) lista(n)={my(o=(1-x^(n+1))/(1-x)*O(y^(n+2)),p=prod(i=1,n,1+o+x*y^(i+1)/(1-y),1/(1-y)));p=subst(serlaplace(p),x,1);Vec(p-1)} \\ _Christian Sievers_, Jun 18 2025
%Y A384175 For equal instead of distinct lengths we have A243815.
%Y A384175 These subsets are ranked by A328592.
%Y A384175 The complement is counted by A384176.
%Y A384175 For anti-runs instead of runs we have A384177, ranks A384879.
%Y A384175 For partitions instead of subsets we have A384884, A384178, A384886, A384880.
%Y A384175 For permutations instead of subsets we have A384891, equal instead of distinct A384892.
%Y A384175 A034839 counts subsets by number of maximal runs, for strict partitions A116674.
%Y A384175 A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
%Y A384175 A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
%Y A384175 Cf. A000009, A010027, A044813, A047993, A242882, A325325, A329739, A351202, A383013, A384889, A384890.
%K A384175 nonn
%O A384175 0,2
%A A384175 _Gus Wiseman_, Jun 16 2025
%E A384175 a(21) and beyond from _Christian Sievers_, Jun 18 2025