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A374635 Number of integer compositions of n whose leaders of weakly increasing runs are themselves weakly increasing.

%I A374635 #11 Feb 14 2025 01:14:06
%S A374635 1,1,2,3,6,10,20,36,69,130,247,467,890,1689,3213,6110,11627,22121,
%T A374635 42101,80124,152512,290300,552609,1051953,2002583,3812326,7257679,
%U A374635 13816867,26304254,50077792,95338234,181505938,345554234,657874081,1252478707,2384507463,4539705261
%N A374635 Number of integer compositions of n whose leaders of weakly increasing runs are themselves weakly increasing.
%C A374635 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 A374635 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374635 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 A374635 The a(0) = 1 through a(6) = 20 compositions:
%e A374635   ()  (1)  (2)   (3)    (4)     (5)      (6)
%e A374635            (11)  (12)   (13)    (14)     (15)
%e A374635                  (111)  (22)    (23)     (24)
%e A374635                         (112)   (113)    (33)
%e A374635                         (121)   (122)    (114)
%e A374635                         (1111)  (131)    (123)
%e A374635                                 (1112)   (132)
%e A374635                                 (1121)   (141)
%e A374635                                 (1211)   (222)
%e A374635                                 (11111)  (1113)
%e A374635                                          (1122)
%e A374635                                          (1131)
%e A374635                                          (1212)
%e A374635                                          (1221)
%e A374635                                          (1311)
%e A374635                                          (11112)
%e A374635                                          (11121)
%e A374635                                          (11211)
%e A374635                                          (12111)
%e A374635                                          (111111)
%t A374635 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],LessEqual@@First/@Split[#,LessEqual]&]],{n,0,15}]
%o A374635 (PARI) dfs(m, r, u) = 1 + sum(s=u, min(m, r-1), x^s/(1-x^s) + sum(t=s+1, m-s, dfs(m-s-t, t, s)*x^(s+t)/prod(i=s, t, 1-x^i)));
%o A374635 lista(nn) = Vec(dfs(nn, nn+1, 1) + O(x^(1+nn))); \\ _Jinyuan Wang_, Feb 13 2025
%Y A374635 Ranked by positions of weakly increasing rows in A374629 (sums A374630).
%Y A374635 Types of runs (instead of weakly increasing):
%Y A374635 - For leaders of constant runs we have A000041.
%Y A374635 - For leaders of weakly decreasing runs we have A188900.
%Y A374635 - For leaders of anti-runs we have A374681.
%Y A374635 - For leaders of strictly increasing runs we have A374690.
%Y A374635 - For leaders of strictly decreasing runs we have A374764.
%Y A374635 Types of run-leaders (instead of weakly increasing):
%Y A374635 - For strictly decreasing leaders we appear to have A188920.
%Y A374635 - For weakly decreasing leaders we appear to have A189076.
%Y A374635 - For identical leaders we have A374631.
%Y A374635 - For distinct leaders we have A374632, ranks A374768.
%Y A374635 - For strictly increasing leaders we have A374634.
%Y A374635 A003242 counts anti-run compositions.
%Y A374635 A011782 counts compositions.
%Y A374635 A238130, A238279, A333755 count compositions by number of runs.
%Y A374635 A274174 counts contiguous compositions, ranks A374249.
%Y A374635 A335456 counts patterns matched by compositions.
%Y A374635 A335548 counts non-contiguous compositions, ranks A374253.
%Y A374635 A374637 counts compositions by sum of leaders of weakly increasing runs.
%Y A374635 Cf. A106356, A124766, A238343, A261982, A333213, A374518, A374687, A374761.
%K A374635 nonn
%O A374635 0,3
%A A374635 _Gus Wiseman_, Jul 23 2024
%E A374635 More terms from _Jinyuan Wang_, Feb 13 2025