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A374747 Number of integer compositions of n whose leaders of weakly decreasing runs are themselves weakly decreasing.

%I A374747 #12 Feb 14 2025 01:17:19
%S A374747 1,1,2,3,5,8,14,24,43,76,136,242,431,764,1353,2387,4202,7376,12918,
%T A374747 22567,39338,68421,118765,205743,355756,614038,1058023,1820029,
%U A374747 3125916,5360659,9179700,15697559,26807303,45720739,77881393,132505599,225182047,382252310,648187055
%N A374747 Number of integer compositions of n whose leaders of weakly decreasing runs are themselves weakly decreasing.
%C A374747 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 A374747 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374747 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 A374747 The a(0) = 1 through a(6) = 14 compositions:
%e A374747   ()  (1)  (2)   (3)    (4)     (5)      (6)
%e A374747            (11)  (21)   (22)    (32)     (33)
%e A374747                  (111)  (31)    (41)     (42)
%e A374747                         (211)   (212)    (51)
%e A374747                         (1111)  (221)    (222)
%e A374747                                 (311)    (312)
%e A374747                                 (2111)   (321)
%e A374747                                 (11111)  (411)
%e A374747                                          (2112)
%e A374747                                          (2121)
%e A374747                                          (2211)
%e A374747                                          (3111)
%e A374747                                          (21111)
%e A374747                                          (111111)
%t A374747 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],GreaterEqual@@First/@Split[#,GreaterEqual]&]],{n,0,15}]
%o A374747 (PARI) dfs(m, r, u) = 1 + sum(s=r+1, min(m, u), x^s/(1-x^s) + sum(t=1, min(s-1, m-s), dfs(m-s-t, t, s)*x^(s+t)/prod(i=t, s, 1-x^i)));
%o A374747 lista(nn) = Vec(dfs(nn, 0, nn) + O(x^(1+nn))); \\ _Jinyuan Wang_, Feb 14 2025
%Y A374747 Ranked by positions of weakly decreasing rows in A374740, opposite A374629.
%Y A374747 Types of runs (instead of weakly decreasing):
%Y A374747 - For leaders of identical runs we have A000041.
%Y A374747 - For leaders of weakly increasing runs we appear to have A189076.
%Y A374747 - For leaders of anti-runs we have A374682.
%Y A374747 - For leaders of strictly increasing runs we have A374697.
%Y A374747 - For leaders of strictly decreasing runs we have A374765.
%Y A374747 Types of run-leaders (instead of weakly decreasing):
%Y A374747 - For weakly increasing leaders we appear to have A188900.
%Y A374747 - For identical leaders we have A374742, ranks A374744.
%Y A374747 - For distinct leaders we have A374743, ranks A374701.
%Y A374747 - For strictly increasing leaders we have opposite A374634.
%Y A374747 - For strictly decreasing leaders we have A374746.
%Y A374747 A011782 counts compositions.
%Y A374747 A124765 counts weakly decreasing runs in standard compositions.
%Y A374747 A238130, A238279, A333755 count compositions by number of runs.
%Y A374747 A335456 counts patterns matched by compositions.
%Y A374747 A373949 counts compositions by run-compressed sum, opposite A373951.
%Y A374747 A374748 counts compositions by sum of leaders of weakly decreasing runs.
%Y A374747 Cf. A000009, A003242, A106356, A188920, A238343, A261982, A333213, A374630, A374635, A374636, A374741.
%K A374747 nonn
%O A374747 0,3
%A A374747 _Gus Wiseman_, Jul 26 2024
%E A374747 More terms from _Jinyuan Wang_, Feb 14 2025