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A374765 Number of integer compositions of n whose leaders of strictly decreasing runs are weakly decreasing.

%I A374765 #11 Feb 14 2025 00:50:37
%S A374765 1,1,2,3,5,8,13,21,34,55,88,141,225,357,565,891,1399,2191,3420,5321,
%T A374765 8256,12774,19711,30339,46584,71359,109066,166340,253163,384539,
%U A374765 582972,882166,1332538,2009377,3024969,4546562,6822926,10223632,15297051,22855872,34103117
%N A374765 Number of integer compositions of n whose leaders of strictly decreasing runs are weakly decreasing.
%C A374765 The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.
%H A374765 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374765 The composition (3,1,2,2,1) has strictly decreasing runs ((3,1),(2),(2,1)), with leaders (3,2,2), so is counted under a(9).
%e A374765 The a(0) = 1 through a(6) = 13 compositions:
%e A374765   ()  (1)  (2)   (3)    (4)     (5)      (6)
%e A374765            (11)  (21)   (22)    (32)     (33)
%e A374765                  (111)  (31)    (41)     (42)
%e A374765                         (211)   (212)    (51)
%e A374765                         (1111)  (221)    (222)
%e A374765                                 (311)    (312)
%e A374765                                 (2111)   (321)
%e A374765                                 (11111)  (411)
%e A374765                                          (2121)
%e A374765                                          (2211)
%e A374765                                          (3111)
%e A374765                                          (21111)
%e A374765                                          (111111)
%t A374765 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],GreaterEqual@@First/@Split[#,Greater]&]],{n,0,15}]
%o A374765 (PARI) dfs(m, r, u) = 1 + sum(s=r, min(m, u), dfs(m-s, s, s)*x^s + sum(t=1, min(s-1, m-s), dfs(m-s-t, t, s)*x^(s+t)*prod(i=t+1, s-1, 1+x^i)));
%o A374765 lista(nn) = Vec(dfs(nn, 1, nn) + O(x^(1+nn))); \\ _Jinyuan Wang_, Feb 13 2025
%Y A374765 The opposite version is A374690.
%Y A374765 Other types of runs (instead of strictly decreasing):
%Y A374765 - For leaders of identical runs we have A000041.
%Y A374765 - For leaders of weakly increasing runs we appear to have A189076.
%Y A374765 - For leaders of anti-runs we have A374682.
%Y A374765 - For leaders of strictly increasing runs we have A374697.
%Y A374765 - For leaders of weakly decreasing runs we have A374747.
%Y A374765 Other types of run-leaders (instead of weakly decreasing):
%Y A374765 - For identical leaders we have A374760, ranks A374759.
%Y A374765 - For distinct leaders we have A374761, ranks A374767.
%Y A374765 - For strictly increasing leaders we have A374762.
%Y A374765 - For strictly decreasing leaders we have A374763.
%Y A374765 - For weakly increasing leaders we have A374764.
%Y A374765 A003242 counts anti-run compositions, ranks A333489.
%Y A374765 A011782 counts compositions.
%Y A374765 A238130, A238279, A333755 count compositions by number of runs.
%Y A374765 A274174 counts contiguous compositions, ranks A374249.
%Y A374765 A373949 counts compositions by run-compressed sum, opposite A373951.
%Y A374765 Cf. A106356, A188900, A188920, A238343, A261982, A333213, A374635, A374636, A374689, A374742, A374743, A375133.
%K A374765 nonn
%O A374765 0,3
%A A374765 _Gus Wiseman_, Jul 30 2024
%E A374765 More terms from _Jinyuan Wang_, Feb 13 2025