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Number of compositions of n that are empty, a singleton, or whose run-lengths are a subsequence that is already counted.
11

%I #6 May 16 2022 17:23:46

%S 1,1,1,1,2,1,3,1,1,4,5,7,9,11,15,22,38,45,87,93

%N Number of compositions of n that are empty, a singleton, or whose run-lengths are a subsequence that is already counted.

%e The a(9) = 4 through a(14) = 15 compositions (A..E = 10..14):

%e (9) (A) (B) (C) (D) (E)

%e (333) (2233) (141122) (2244) (161122) (2255)

%e (121122) (3322) (221123) (4422) (221125) (5522)

%e (221121) (131122) (221132) (151122) (221134) (171122)

%e (221131) (221141) (221124) (221143) (221126)

%e (231122) (221142) (221152) (221135)

%e (321122) (221151) (221161) (221153)

%e (241122) (251122) (221162)

%e (421122) (341122) (221171)

%e (431122) (261122)

%e (521122) (351122)

%e (531122)

%e (621122)

%e (122121122)

%e (221121221)

%t yosQ[y_]:=Length[y]<=1||MemberQ[Subsets[y],Length/@Split[y]]&&yosQ[Length/@Split[y]];

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],yosQ]],{n,0,15}]

%Y The non-recursive version is A353390, ranked by A353402.

%Y The non-recursive consecutive version is A353392, ranked by A353432.

%Y The non-recursive reverse version is A353403.

%Y The unordered version is A353426, ranked by A353393 (nonprime A353389).

%Y The consecutive version is A353430.

%Y These compositions are ranked by A353431.

%Y A003242 counts anti-run compositions, ranked by A333489.

%Y A011782 counts compositions.

%Y A329738 counts uniform compositions, partitions A047966.

%Y A114901 counts compositions with no runs of length 1.

%Y A169942 counts Golomb rulers, ranked by A333222.

%Y A325676 counts knapsack compositions, ranked by A333223.

%Y A325705 counts partitions containing all of their distinct multiplicities.

%Y A329739 counts compositions with all distinct run-length.

%Y Cf. A005811, A032020, A103295, A114640, A165413, A181591, A242882, A324572, A325702, A333755, A351013, A353401.

%K nonn,more

%O 0,5

%A _Gus Wiseman_, May 15 2022