cp's OEIS Frontend

A375140 Number of integer compositions of n whose leaders of weakly increasing runs are not strictly decreasing.

%I A375140 #8 Aug 13 2024 09:09:41
%S A375140 0,0,0,1,3,10,26,65,151,343,750,1614,3410,7123,14724,30220,61639,
%T A375140 125166,253233,510936,1028659,2067620,4150699,8324552,16683501,
%U A375140 33417933,66910805,133931495,268023257,536279457,1072895973,2146277961,4293254010,8587507415
%N A375140 Number of integer compositions of n whose leaders of weakly increasing runs are not strictly decreasing.
%C A375140 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.
%C A375140 Also the number of integer compositions of n matching the dashed patterns 1-32 or 1-21.
%H A375140 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>.
%H A375140 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A375140 a(n) = 2^(n-1) - A188920(n).
%e A375140 The a(1) = 0 through a(6) = 10 compositions:
%e A375140      .  .  .  (121)  (131)   (132)
%e A375140                      (1121)  (141)
%e A375140                      (1211)  (1131)
%e A375140                              (1212)
%e A375140                              (1221)
%e A375140                              (1311)
%e A375140                              (2121)
%e A375140                              (11121)
%e A375140                              (11211)
%e A375140                              (12111)
%t A375140 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],!Greater@@First/@Split[#,LessEqual]&]],{n,15}]
%t A375140 - or -
%t A375140 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],MatchQ[#,{___,x_,___,z_,y_,___}/;x<=y<z]&]],{n,0,15}]
%Y A375140 For leaders of identical runs we have A056823.
%Y A375140 The complement is counted by A188920.
%Y A375140 Leaders of weakly increasing runs are rows of A374629, sum A374630.
%Y A375140 For weakly decreasing leaders we have A374636, ranks A375137 or A375138.
%Y A375140 For leaders of weakly decreasing runs we have the complement of A374746.
%Y A375140 Compositions of this type are ranked by A375295, reverse A375296.
%Y A375140 A003242 counts anti-runs, ranks A333489.
%Y A375140 A106356 counts compositions by number of maximal anti-runs.
%Y A375140 A238424 counts partitions whose first differences are an anti-run.
%Y A375140 A274174 counts contiguous compositions, ranks A374249.
%Y A375140 A011782 counts compositions.
%Y A375140 A238130, A238279, A333755 count compositions by number of runs.
%Y A375140 A335456 counts patterns matched by compositions.
%Y A375140 A335548 counts non-contiguous compositions, ranks A374253.
%Y A375140 A374637 counts compositions by sum of leaders of weakly increasing runs.
%Y A375140 Cf. A124766, A238343, A261982, A333213, A335514, A373949, A374635, A374678, A374687, A374761.
%K A375140 nonn
%O A375140 1,5
%A A375140 _Gus Wiseman_, Aug 10 2024