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A384177 Number of subsets of {1..n} with all distinct lengths of maximal anti-runs (increasing by more than 1).

%I A384177 #13 Jun 18 2025 23:18:43
%S A384177 1,2,3,5,10,19,35,62,109,197,364,677,1251,2288,4143,7443,13318,23837,
%T A384177 42809,77216,139751,253293,458800,829237,1494169,2683316,4804083,
%U A384177 8580293,15301324,27270061,48607667,86696300,154758265,276453311,494050894,882923051
%N A384177 Number of subsets of {1..n} with all distinct lengths of maximal anti-runs (increasing by more than 1).
%H A384177 Christian Sievers, <a href="/A384177/b384177.txt">Table of n, a(n) for n = 0..1000</a>
%e A384177 The subset {1,2,4,5,7,10} has maximal anti-runs ((1),(2,4),(5,7,10)), with lengths (1,2,3), so is counted under a(10).
%e A384177 The a(0) = 1 through a(5) = 19 subsets:
%e A384177   {}  {}   {}   {}     {}       {}
%e A384177       {1}  {1}  {1}    {1}      {1}
%e A384177            {2}  {2}    {2}      {2}
%e A384177                 {3}    {3}      {3}
%e A384177                 {1,3}  {4}      {4}
%e A384177                        {1,3}    {5}
%e A384177                        {1,4}    {1,3}
%e A384177                        {2,4}    {1,4}
%e A384177                        {1,2,4}  {1,5}
%e A384177                        {1,3,4}  {2,4}
%e A384177                                 {2,5}
%e A384177                                 {3,5}
%e A384177                                 {1,2,4}
%e A384177                                 {1,2,5}
%e A384177                                 {1,3,4}
%e A384177                                 {1,3,5}
%e A384177                                 {1,4,5}
%e A384177                                 {2,3,5}
%e A384177                                 {2,4,5}
%t A384177 Table[Length[Select[Subsets[Range[n]],UnsameQ@@Length/@Split[#,#2!=#1+1&]&]],{n,0,10}]
%o A384177 (PARI) lista(n)={my(o=(1-x^(n+1))/(1-x)*O(y*y^n),p=prod(i=1,(n+1)\2,1+o+x*y^(2*i-1)/(1-y)^(i-1)));p=subst(serlaplace(p),x,1);Vec((p-y)/(1-y)^2)} \\ _Christian Sievers_, Jun 18 2025
%Y A384177 For runs instead of anti-runs we have A384175, complement A384176.
%Y A384177 These subsets are ranked by A384879.
%Y A384177 For strict partitions instead of subsets we have A384880, see A384178, A384884, A384886.
%Y A384177 For equal instead of distinct lengths we have A384889, for runs A243815.
%Y A384177 A034839 counts subsets by number of maximal runs, for strict partitions A116674.
%Y A384177 A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
%Y A384177 A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
%Y A384177 Cf. A000009, A010027, A044813, A047993, A106529, A123513, A242882, A325325, A328592, A329739, A351202, A384890.
%K A384177 nonn
%O A384177 0,2
%A A384177 _Gus Wiseman_, Jun 16 2025
%E A384177 a(21) and beyond from _Christian Sievers_, Jun 18 2025