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%I #6 May 16 2022 10:02:31
%S 1,1,0,0,1,2,3,2,2,8,17,26,43,77,129,210,351,569
%N Number of compositions of n whose own run-lengths are a subsequence (not necessarily consecutive).
%e The a(0) = 1 through a(9) = 8 compositions (empty columns indicated by dots):
%e () (1) . . (22) (122) (1122) (11221) (21122) (333)
%e (221) (1221) (12211) (22112) (22113)
%e (2211) (22122)
%e (31122)
%e (121122)
%e (122112)
%e (211221)
%e (221121)
%e For example, the composition y = (2,2,3,3,1) has run-lengths (2,2,1), which form a (non-consecutive) subsequence, so y is counted under a(11).
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], MemberQ[Subsets[#],Length/@Split[#]]&]],{n,0,15}]
%Y The version for partitions is A325702.
%Y The recursive version is A353391, ranked by A353431.
%Y The consecutive case is A353392, ranked by A353432.
%Y These compositions are ranked by A353402.
%Y The reverse version is A353403.
%Y The recursive consecutive version is A353430.
%Y A003242 counts anti-run compositions, ranked by A333489.
%Y A011782 counts compositions.
%Y A047966 counts uniform partitions, compositions A329738.
%Y A169942 counts Golomb rulers, ranked by A333222.
%Y A325676 counts knapsack compositions, ranked by A333223, partitions A108917.
%Y A325705 counts partitions containing all of their distinct multiplicities.
%Y A329739 counts compositions with all distinct run-lengths, for runs A351013.
%Y A353400 counts compositions with all run-lengths > 2.
%Y Cf. A005811, A103295, A114901, A181591, A238279, A242882, A324572, A333755, A351017, A353401, A353426.
%K nonn,more
%O 0,6
%A _Gus Wiseman_, May 15 2022
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