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A374634 Number of integer compositions of n whose leaders of weakly increasing runs are strictly increasing.

%I A374634 #13 Feb 14 2025 01:14:00
%S A374634 1,1,2,3,5,7,12,17,28,43,67,103,162,245,374,569,854,1278,1902,2816,
%T A374634 4148,6087,8881,12926,18726,27042,38894,55789,79733,113632,161426,
%U A374634 228696,323049,455135,639479,896249,1252905,1747327,2431035,3374603,4673880,6459435,8908173
%N A374634 Number of integer compositions of n whose leaders of weakly increasing runs are strictly increasing.
%C A374634 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 A374634 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374634 The composition (1,3,3,2,4,3) has weakly increasing runs ((1,3,3),(2,4),(3)), with leaders (1,2,3), so is counted under a(16).
%e A374634 The a(0) = 1 through a(7) = 17 compositions:
%e A374634   ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)
%e A374634            (11)  (12)   (13)    (14)     (15)      (16)
%e A374634                  (111)  (22)    (23)     (24)      (25)
%e A374634                         (112)   (113)    (33)      (34)
%e A374634                         (1111)  (122)    (114)     (115)
%e A374634                                 (1112)   (123)     (124)
%e A374634                                 (11111)  (132)     (133)
%e A374634                                          (222)     (142)
%e A374634                                          (1113)    (223)
%e A374634                                          (1122)    (1114)
%e A374634                                          (11112)   (1123)
%e A374634                                          (111111)  (1132)
%e A374634                                                    (1222)
%e A374634                                                    (11113)
%e A374634                                                    (11122)
%e A374634                                                    (111112)
%e A374634                                                    (1111111)
%t A374634 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Less@@First/@Split[#,LessEqual]&]],{n,0,15}]
%o A374634 (PARI) dfs(m, r, u) = 1 + sum(s=u+1, 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 A374634 lista(nn) = Vec(dfs(nn, nn+1, 0) + O(x^(1+nn))); \\ _Jinyuan Wang_, Feb 13 2025
%Y A374634 Ranked by positions of strictly increasing rows in A374629 (sums A374630).
%Y A374634 Types of runs (instead of weakly increasing):
%Y A374634 - For leaders of constant runs we have A000041.
%Y A374634 - For leaders of anti-runs we have A374679.
%Y A374634 - For leaders of strictly increasing runs we have A374688.
%Y A374634 - For leaders of strictly decreasing runs we have A374762.
%Y A374634 Types of run-leaders (instead of strictly increasing):
%Y A374634 - For strictly decreasing leaders we appear to have A188920.
%Y A374634 - For weakly decreasing leaders we appear to have A189076.
%Y A374634 - For identical leaders we have A374631.
%Y A374634 - For distinct leaders we have A374632, ranks A374768.
%Y A374634 - For weakly increasing leaders we have A374635.
%Y A374634 A003242 counts anti-run compositions.
%Y A374634 A011782 counts compositions.
%Y A374634 A238130, A238279, A333755 count compositions by number of runs.
%Y A374634 A274174 counts contiguous compositions, ranks A374249.
%Y A374634 A335456 counts patterns matched by compositions.
%Y A374634 A335548 counts non-contiguous compositions, ranks A374253.
%Y A374634 A374637 counts compositions by sum of leaders of weakly increasing runs.
%Y A374634 Cf. A106356, A124766, A238343, A261982, A333213, A344604, A373949, A374518, A374687, A374698.
%K A374634 nonn
%O A374634 0,3
%A A374634 _Gus Wiseman_, Jul 23 2024
%E A374634 More terms from _Jinyuan Wang_, Feb 13 2025