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A385214 Number of subsets of {1..n} without all equal lengths of maximal runs of consecutive elements increasing by 1.

%I A385214 #5 Jun 25 2025 18:04:35
%S A385214 0,0,0,0,2,8,25,66,159,361,791,1688,3539,7328,15040,30669,62246,
%T A385214 125896,253975,511357,1028052
%N A385214 Number of subsets of {1..n} without all equal lengths of maximal runs of consecutive elements increasing by 1.
%e A385214 The maximal runs of S = {1,2,4,5,6,8,9} are ((1,2),(4,5,6),(8,9)), with lengths (2,3,2), so S is counted under a(9).
%e A385214 The a(0) = 0 through a(5) = 8 subsets:
%e A385214   .  .  .  .  {1,2,4}  {1,2,4}
%e A385214               {1,3,4}  {1,2,5}
%e A385214                        {1,3,4}
%e A385214                        {1,4,5}
%e A385214                        {2,3,5}
%e A385214                        {2,4,5}
%e A385214                        {1,2,3,5}
%e A385214                        {1,3,4,5}
%t A385214 Table[Length[Select[Subsets[Range[n]],!SameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}]
%Y A385214 These subsets are ranked by A164708, complement A164707
%Y A385214 The complement is counted by A243815.
%Y A385214 For distinct instead of equal lengths we have A384176, complement A384175.
%Y A385214 For anti-runs instead of runs we have complement of A384889, for partitions A384888.
%Y A385214 For permutations instead of subsets we have complement of A384892, distinct A384891.
%Y A385214 For partitions instead of subsets we have complement of A384904, strict A384886.
%Y A385214 A034839 counts subsets by number of maximal runs, for strict partitions A116674.
%Y A385214 A049988 counts partitions with equal run-lengths, distinct A325325.
%Y A385214 A329738 counts compositions with equal run-lengths, distinct A329739.
%Y A385214 A384177 counts subsets with all distinct lengths of maximal anti-runs, ranks A384879.
%Y A385214 A384887 counts partitions with equal lengths of gapless runs, distinct A384884.
%Y A385214 A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
%Y A385214 Cf. A010027, A044813, A164710, A383013, A384885.
%K A385214 nonn,more
%O A385214 0,5
%A A385214 _Gus Wiseman_, Jun 25 2025