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A384889 Number of subsets of {1..n} with all equal lengths of maximal anti-runs (increasing by more than 1).

%I A384889 #15 Jun 22 2025 15:08:25
%S A384889 1,2,4,8,14,23,37,59,93,146,230,365,584,940,1517,2450,3959,6404,10373,
%T A384889 16822,27298,44297,71843,116429,188550,305200,493930,799422,1294108,
%U A384889 2095291,3392736,5493168,8892148,14390372,23282110,37660759,60914308,98528312,159386110
%N A384889 Number of subsets of {1..n} with all equal lengths of maximal anti-runs (increasing by more than 1).
%H A384889 Christian Sievers, <a href="/A384889/b384889.txt">Table of n, a(n) for n = 0..1000</a>
%F A384889 G.f.: ( Sum_{i>=1} (1/(1-x^(2*i-1)/(1-x)^(i-1))-1) + 1-x ) / (1-x)^2. - _Christian Sievers_, Jun 21 2025
%e A384889 The subset {3,6,7,9,10,12} has maximal anti-runs ((3,6),(7,9),(10,12)), with lengths (2,2,2), so is counted under a(12).
%e A384889 The a(0) = 1 through a(4) = 14 subsets:
%e A384889   {}  {}   {}     {}       {}
%e A384889       {1}  {1}    {1}      {1}
%e A384889            {2}    {2}      {2}
%e A384889            {1,2}  {3}      {3}
%e A384889                   {1,2}    {4}
%e A384889                   {1,3}    {1,2}
%e A384889                   {2,3}    {1,3}
%e A384889                   {1,2,3}  {1,4}
%e A384889                            {2,3}
%e A384889                            {2,4}
%e A384889                            {3,4}
%e A384889                            {1,2,3}
%e A384889                            {2,3,4}
%e A384889                            {1,2,3,4}
%t A384889 Table[Length[Select[Subsets[Range[n]],SameQ@@Length/@Split[#,#2!=#1+1&]&]],{n,0,10}]
%o A384889 (PARI) lista(n)=Vec(sum(i=1,(n+1)\2,1/(1-x^(2*i-1)/(1-x)^(i-1))-1,1-x+O(x*x^n))/(1-x)^2) \\ _Christian Sievers_, Jun 20 2025
%Y A384889 For runs instead of anti-runs we have A243815, distinct A384175, complement A384176.
%Y A384889 For distinct instead or equal lengths we have A384177, ranks A384879.
%Y A384889 For partitions instead of subsets we have A384888.
%Y A384889 A034296 counts flat or gapless partitions, ranks A066311 or A073491.
%Y A384889 A034839 counts subsets by number of maximal runs, for strict partitions A116674.
%Y A384889 A047966 counts uniform partitions (equal multiplicities), ranks A072774.
%Y A384889 A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
%Y A384889 Cf. A010027, A044813, A047993, A356606, A384880, A384886, A384890.
%K A384889 nonn
%O A384889 0,2
%A A384889 _Gus Wiseman_, Jun 18 2025
%E A384889 a(21) and beyond from _Christian Sievers_, Jun 20 2025