cp's OEIS Frontend

A375135 Number of integer compositions of n whose leaders of maximal strictly increasing runs are not weakly decreasing.

%I A375135 #11 Feb 14 2025 01:13:44
%S A375135 0,0,0,0,0,1,3,9,25,63,152,355,809,1804,3963,8590,18423,39161,82620,
%T A375135 173198,361101,749326,1548609,3189132,6547190,13404613,27378579,
%U A375135 55801506,113517749,230544752,467519136,946815630,1915199736,3869892105,7812086380,15756526347
%N A375135 Number of integer compositions of n whose leaders of maximal strictly increasing runs are not weakly decreasing.
%C A375135 The leaders of maximal strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.
%H A375135 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A375135 a(n) = A011782(n) - A374697(n). - _Jinyuan Wang_, Feb 13 2025
%e A375135 The composition y = (1,2,1,3,2,3) has strictly increasing runs ((1,2),(1,3),(2,3)), with leaders (1,1,2), which are not weakly decreasing, so y is counted under a(12).
%e A375135 The a(0) = 0 through a(8) = 25 compositions:
%e A375135   .  .  .  .  .  (122)  (132)   (133)    (143)
%e A375135                         (1122)  (142)    (152)
%e A375135                         (1221)  (1132)   (233)
%e A375135                                 (1222)   (1133)
%e A375135                                 (1321)   (1142)
%e A375135                                 (2122)   (1223)
%e A375135                                 (11122)  (1232)
%e A375135                                 (11221)  (1322)
%e A375135                                 (12211)  (1331)
%e A375135                                          (1421)
%e A375135                                          (2132)
%e A375135                                          (3122)
%e A375135                                          (11132)
%e A375135                                          (11222)
%e A375135                                          (11321)
%e A375135                                          (12122)
%e A375135                                          (12212)
%e A375135                                          (12221)
%e A375135                                          (13211)
%e A375135                                          (21122)
%e A375135                                          (21221)
%e A375135                                          (111122)
%e A375135                                          (111221)
%e A375135                                          (112211)
%e A375135                                          (122111)
%t A375135 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], !GreaterEqual@@First/@Split[#,Less]&]],{n,0,15}]
%Y A375135 For leaders of constant runs we have A056823.
%Y A375135 For leaders of weakly increasing runs we have A374636, complement A189076?
%Y A375135 The complement is counted by A374697.
%Y A375135 For leaders of anti-runs we have A374699, complement A374682.
%Y A375135 Other functional neighbors: A188920, A374764, A374765.
%Y A375135 A003242 counts anti-run compositions, ranks A333489.
%Y A375135 A011782 counts compositions.
%Y A375135 A238130, A238279, A333755 count compositions by number of runs.
%Y A375135 A335456 counts patterns matched by compositions.
%Y A375135 A373949 counts compositions by run-compressed sum, opposite A373951.
%Y A375135 A374700 counts compositions by sum of leaders of strictly increasing runs.
%Y A375135 Cf. A106356, A238343, A261982, A333213, A374632, A374679, A374683, A374689.
%K A375135 nonn
%O A375135 0,7
%A A375135 _Gus Wiseman_, Aug 06 2024
%E A375135 More terms from _Jinyuan Wang_, Feb 13 2025