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A374743 Number of integer compositions of n whose leaders of weakly decreasing runs are distinct.

%I A374743 #11 Sep 16 2024 08:42:23
%S A374743 1,1,2,4,8,15,29,55,105,198,371,690,1280,2364,4353,7981,14568,26466,
%T A374743 47876,86264,154896,277236,494675,879924,1560275,2757830,4859010,
%U A374743 8534420,14945107,26096824,45446624,78939432,136773519,236401194,407614349,701147189,1203194421
%N A374743 Number of integer compositions of n whose leaders of weakly decreasing runs are distinct.
%C A374743 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 A374743 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374743 The composition (1,3,1,4,1,2,2,1) has maximal weakly decreasing subsequences ((1),(3,1),(4,1),(2,2,1)), with leaders (1,3,4,2), so is counted under a(15).
%e A374743 The a(0) = 1 through a(5) = 15 compositions:
%e A374743   ()  (1)  (2)   (3)    (4)     (5)
%e A374743            (11)  (12)   (13)    (14)
%e A374743                  (21)   (22)    (23)
%e A374743                  (111)  (31)    (32)
%e A374743                         (112)   (41)
%e A374743                         (121)   (113)
%e A374743                         (211)   (122)
%e A374743                         (1111)  (131)
%e A374743                                 (221)
%e A374743                                 (311)
%e A374743                                 (1112)
%e A374743                                 (1121)
%e A374743                                 (1211)
%e A374743                                 (2111)
%e A374743                                 (11111)
%t A374743 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],UnsameQ@@First/@Split[#,GreaterEqual]&]],{n,0,15}]
%Y A374743 Ranked by A374701 = positions of distinct rows in A374740, opposite A374629.
%Y A374743 Types of runs (instead of weakly decreasing):
%Y A374743 - For leaders of identical runs we have A274174, ranks A374249.
%Y A374743 - For leaders of anti-runs we have A374518, ranks A374638.
%Y A374743 - For leaders of weakly increasing runs we have A374632, ranks A374768.
%Y A374743 - For leaders of strictly increasing runs we have A374687, ranks A374698.
%Y A374743 - For leaders of strictly decreasing runs we have A374761, ranks A374767.
%Y A374743 Types of run-leaders (instead of distinct):
%Y A374743 - For weakly increasing leaders we appear to have A188900.
%Y A374743 - For identical leaders we have A374742.
%Y A374743 - For strictly increasing leaders we have opposite A374634.
%Y A374743 - For strictly decreasing leaders we have A374746.
%Y A374743 - For weakly decreasing leaders we have A374747.
%Y A374743 A011782 counts compositions.
%Y A374743 A238130, A238279, A333755 count compositions by number of runs.
%Y A374743 A335456 counts patterns matched by compositions.
%Y A374743 A373949 counts compositions by run-compressed sum, opposite A373951.
%Y A374743 A374748 counts compositions by sum of leaders of weakly decreasing runs.
%Y A374743 Cf. A000009, A003242, A106356, A188920, A189076, A238343, A261982, A333213, A374635, A374741.
%K A374743 nonn
%O A374743 0,3
%A A374743 _Gus Wiseman_, Jul 25 2024
%E A374743 a(24)-a(36) from _Alois P. Heinz_, Jul 26 2024