%I #7 Jul 24 2024 09:21:12
%S 1,1,2,3,6,10,20,36,69,130,247,467,890,1689,3213,6110,11627,22121,
%T 42101,80124,152512,290300,552609,1051953,2002583,3812326
%N Number of integer compositions of n whose leaders of weakly increasing runs are themselves weakly increasing.
%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 (1,3,3,2,4,2) has weakly increasing runs ((1,3,3),(2,4),(2)), with leaders (1,2,2), so is counted under a(15).
%e The a(0) = 1 through a(6) = 20 compositions:
%e () (1) (2) (3) (4) (5) (6)
%e (11) (12) (13) (14) (15)
%e (111) (22) (23) (24)
%e (112) (113) (33)
%e (121) (122) (114)
%e (1111) (131) (123)
%e (1112) (132)
%e (1121) (141)
%e (1211) (222)
%e (11111) (1113)
%e (1122)
%e (1131)
%e (1212)
%e (1221)
%e (1311)
%e (11112)
%e (11121)
%e (11211)
%e (12111)
%e (111111)
%t Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],LessEqual@@First/@Split[#,LessEqual]&]],{n,0,15}]
%Y Ranked by positions of weakly increasing rows in A374629 (sums A374630).
%Y Types of runs (instead of weakly increasing):
%Y - For leaders of constant runs we have A000041.
%Y - For leaders of weakly decreasing runs we have A188900.
%Y - For leaders of anti-runs we have A374681.
%Y - For leaders of strictly increasing runs we have A374690.
%Y - For leaders of strictly decreasing runs we have A374764.
%Y Types of run-leaders (instead of weakly increasing):
%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 distinct leaders we have A374632, ranks A374768.
%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 A274174 counts contiguous compositions, ranks A374249.
%Y A335456 counts patterns matched by compositions.
%Y A335548 counts non-contiguous compositions, ranks A374253.
%Y A374637 counts compositions by sum of leaders of weakly increasing runs.
%Y Cf. A106356, A124766, A238343, A261982, A333213, A374518, A374687, A374761.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Jul 23 2024