A374747 - OEIS

A374747

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

%I #8 Sep 16 2024 08:43:47

%S 1,1,2,3,5,8,14,24,43,76,136,242,431,764,1353,2387,4202,7376,12918,

%T 22567,39338,68421,118765,205743

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

%C The weakly decreasing run-leaders of a sequence are obtained by splitting it into maximal weakly 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 y = (3,2,1,2,2,1,2,5,1,1,1) has weakly decreasing runs ((3,2,1),(2,2,1),(2),(5,1,1,1)), with leaders (3,2,2,5), which are not weakly decreasing, so y is not counted under a(21).

%e The a(0) = 1 through a(6) = 14 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 (2112)

%e (2121)

%e (2211)

%e (3111)

%e (21111)

%e (111111)

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

%Y Ranked by positions of weakly decreasing rows in A374740, opposite A374629.

%Y Types of runs (instead of weakly 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 strictly decreasing runs we have A374765.

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

%Y - For weakly increasing leaders we appear to have A188900.

%Y - For identical leaders we have A374742, ranks A374744.

%Y - For distinct leaders we have A374743, ranks A374701.

%Y - For strictly increasing leaders we have opposite A374634.

%Y - For strictly decreasing leaders we have A374746.

%Y A011782 counts compositions.

%Y A124765 counts weakly decreasing runs in standard compositions.

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

%Y A335456 counts patterns matched by compositions.

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

%Y A374748 counts compositions by sum of leaders of weakly decreasing runs.

%Y Cf. A000009, A003242, A106356, A188920, A238343, A261982, A333213, A374630, A374635, A374636, A374741.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Jul 26 2024