%I #5 Jul 24 2024 09:22:23

%S 1,1,2,4,7,13,23,40,69,119,200,335,557,917,1499,2433,3920,6280,10004,

%T 15837,24946,39087,60952,94606,146203,224957

%N Number of integer compositions of n whose leaders of weakly increasing runs are distinct.

%C The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing 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 (4,2,2,1,1,3) has weakly increasing runs ((4),(2,2),(1,1,3)), with leaders (4,2,1), so is counted under a(13).

%e The a(0) = 1 through a(5) = 13 compositions:

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

%e (11) (12) (13) (14)

%e (21) (22) (23)

%e (111) (31) (32)

%e (112) (41)

%e (211) (113)

%e (1111) (122)

%e (212)

%e (221)

%e (311)

%e (1112)

%e (2111)

%e (11111)

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

%Y Ranked by A374768 = positions of distinct rows in A374629 (sums A374630).

%Y Types of runs (instead of weakly increasing):

%Y - For leaders of constant runs we have A274174, ranks A374249.

%Y - For leaders of anti-runs we have A374518, ranks A374638.

%Y - For leaders of strictly increasing runs we have A374687, ranks A374698.

%Y - For leaders of weakly decreasing runs we have A374743, ranks A335467.

%Y - For leaders of strictly decreasing runs we have A374761, ranks A374767.

%Y Types of run-leaders (instead of distinct):

%Y - For strictly decreasing leaders we appear to have A188920.

%Y - For weakly decreasing leaders we appear to have A189076.

%Y - For identical leaders we have A374631.

%Y - For weakly increasing leaders we have A374635.

%Y - For strictly increasing leaders we have A374634.

%Y A003242 counts anti-run compositions.

%Y A011782 counts compositions.

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

%Y A335456 counts patterns matched by compositions.

%Y A374637 counts compositions by sum of leaders of weakly increasing runs.

%Y Cf. A106356, A124766, A238343, A261982, A333213, A335548, A373949, A373953.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Jul 23 2024