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Number of integer compositions of n whose leaders of strictly decreasing runs are weakly decreasing.
9

%I #7 Jul 31 2024 17:27:42

%S 1,1,2,3,5,8,13,21,34,55,88,141,225,357,565,891,1399,2191,3420,5321,

%T 8256,12774,19711,30339

%N Number of integer compositions of n whose leaders of strictly decreasing runs are weakly decreasing.

%C The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.

%H Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.

%e The composition (3,1,2,2,1) has strictly decreasing runs ((3,1),(2),(2,1)), with leaders (3,2,2), so is counted under a(9).

%e The a(0) = 1 through a(6) = 13 compositions:

%e () (1) (2) (3) (4) (5) (6)

%e (11) (21) (22) (32) (33)

%e (111) (31) (41) (42)

%e (211) (212) (51)

%e (1111) (221) (222)

%e (311) (312)

%e (2111) (321)

%e (11111) (411)

%e (2121)

%e (2211)

%e (3111)

%e (21111)

%e (111111)

%t Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],GreaterEqual@@First/@Split[#,Greater]&]],{n,0,15}]

%Y The opposite version is A374690.

%Y Other types of runs (instead of strictly decreasing):

%Y - For leaders of identical runs we have A000041.

%Y - For leaders of weakly increasing runs we appear to have A189076.

%Y - For leaders of anti-runs we have A374682.

%Y - For leaders of strictly increasing runs we have A374697.

%Y - For leaders of weakly decreasing runs we have A374747.

%Y Other types of run-leaders (instead of weakly decreasing):

%Y - For identical leaders we have A374760, ranks A374759.

%Y - For distinct leaders we have A374761, ranks A374767.

%Y - For strictly increasing leaders we have A374762.

%Y - For strictly decreasing leaders we have A374763.

%Y - For weakly increasing leaders we have A374764.

%Y A003242 counts anti-run compositions, ranks A333489.

%Y A011782 counts compositions.

%Y A238130, A238279, A333755 count compositions by number of runs.

%Y A274174 counts contiguous compositions, ranks A374249.

%Y A373949 counts compositions by run-compressed sum, opposite A373951.

%Y Cf. A106356, A188900, A188920, A238343, A261982, A333213, A374635, A374636, A374689, A374742, A374743, A375133.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Jul 30 2024