This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A374636 #15 Feb 14 2025 01:14:14
%S A374636 0,0,0,0,0,0,1,3,10,28,72,178,425,985,2237,4999,11016,24006,51822,
%T A374636 110983,236064,499168,1050118,2199304,4587946,9537506,19765213,
%U A374636 40847186,84205453,173198096,355520217,728426569,1489977348,3043054678,6206298312,12641504738
%N A374636 Number of integer compositions of n whose leaders of maximal weakly increasing runs are not weakly decreasing.
%C A374636 The leaders of maximal weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.
%C A374636 Also the number of integer compositions of n matching the dashed pattern 1-32, ranked by A375137.
%C A374636 Also the number of integer compositions of n matching the dashed pattern 23-1, ranked by A375138.
%H A374636 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>.
%H A374636 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A374636 a(n) = A011782(n) - A189076(n). - _Jinyuan Wang_, Feb 14 2025
%e A374636 - The maximal weakly increasing runs of y = (1,1,3,2,1) are ((1,1,3),(2),(1)) with leaders (1,2,1) so y is counted under a(8). Also, y matches 1-32 and avoids 23-1.
%e A374636 - The maximal weakly increasing runs of y = (1,3,2,1,1) are ((1,3),(2),(1,1)) with leaders (1,2,1) so y is counted under a(8). Also, y matches 1-32 and avoids 23-1.
%e A374636 - The maximal weakly increasing runs of y = (2,3,1,1,1) are ((2,3),(1,1,1)) with leaders (2,1) so y is not counted under a(8). Also, y avoids 1-32 and matches 23-1.
%e A374636 - The maximal weakly increasing runs of y = (2,3,2,1) are ((2,3),(2),(1)) with leaders (2,2,1) so y is not counted under a(8). Also, y avoids 1-32 and matches 23-1.
%e A374636 - The maximal weakly increasing runs of y = (2,1,3,1,1) are ((2),(1,3),(1,1)) with leaders (2,1,1) so y is not counted under a(8). Also, y avoids both 1-32 and 23-1.
%e A374636 - The maximal weakly increasing runs of y = (2,1,1,3,1) are ((2),(1,1,3),(1)) with leaders (2,1,1) so y is not counted under a(8). Also, y avoids both 1-32 and 23-1.
%e A374636 The a(0) = 0 through a(8) = 10 compositions:
%e A374636 . . . . . . (132) (142) (143)
%e A374636 (1132) (152)
%e A374636 (1321) (1142)
%e A374636 (1232)
%e A374636 (1322)
%e A374636 (1421)
%e A374636 (2132)
%e A374636 (11132)
%e A374636 (11321)
%e A374636 (13211)
%t A374636 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],!GreaterEqual@@First/@Split[#,LessEqual]&]],{n,0,15}]
%t A374636 (* or *)
%t A374636 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],MatchQ[#,{___,y_,z_,___,x_,___}/;x<y<z]&]],{n,0,15}]
%Y A374636 The reverse version is the same.
%Y A374636 For leaders of identical runs we have A056823.
%Y A374636 The complement is counted by A189076.
%Y A374636 The non-dashed version is A335514.
%Y A374636 For leaders of anti-runs we have A374699, complement A374682.
%Y A374636 For weakly decreasing runs we have the complement of A374747.
%Y A374636 For leaders of strictly increasing runs we have A375135, complement A374697.
%Y A374636 These compositions are ranked by A375137, reverse A375138.
%Y A374636 A003242 counts anti-runs, ranks A333489.
%Y A374636 A106356 counts compositions by number of maximal anti-runs.
%Y A374636 A238279 counts compositions by number of maximal runs
%Y A374636 A274174 counts contiguous compositions, ranks A374249.
%Y A374636 Cf. A000041, A188920, A238343, A238424, A333213, A373949, A374632, A374635, A374678, A374681, A375297.
%K A374636 nonn
%O A374636 0,8
%A A374636 _Gus Wiseman_, Aug 09 2024
%E A374636 More terms from _Jinyuan Wang_, Feb 14 2025