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A374699 Number of integer compositions of n whose leaders of maximal anti-runs are not weakly decreasing.

%I A374699 #5 Aug 06 2024 21:37:10
%S A374699 0,0,0,0,0,1,2,5,14,34,78,180,407,907,2000,4364,9448,20323,43448,
%T A374699 92400,195604,412355,866085,1813035,3783895,7875552
%N A374699 Number of integer compositions of n whose leaders of maximal anti-runs are not weakly decreasing.
%C A374699 The leaders of maximal anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.
%H A374699 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374699 The a(0) = 0 through a(8) = 14 compositions:
%e A374699   .  .  .  .  .  (122)  (1122)  (133)    (233)
%e A374699                         (1221)  (1222)   (1133)
%e A374699                                 (11122)  (1223)
%e A374699                                 (11221)  (1322)
%e A374699                                 (12211)  (1331)
%e A374699                                          (11222)
%e A374699                                          (12122)
%e A374699                                          (12212)
%e A374699                                          (12221)
%e A374699                                          (21122)
%e A374699                                          (111122)
%e A374699                                          (111221)
%e A374699                                          (112211)
%e A374699                                          (122111)
%t A374699 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],!GreaterEqual@@First/@Split[#,UnsameQ]&]],{n,0,15}]
%Y A374699 The complement is counted by A374682.
%Y A374699 Other types of runs (instead of anti-):
%Y A374699 - For leaders of identical runs we have A056823.
%Y A374699 - For leaders of weakly increasing runs we have A374636, complement A189076?
%Y A374699 - For leaders of strictly increasing runs: A375135, complement A374697.
%Y A374699 Other types of run-leaders (instead of weakly decreasing):
%Y A374699 - For identical leaders we have A374640, ranks A374520, complement A374517, ranks A374519.
%Y A374699 - For distinct leaders we have A374678, ranks A374639, complement A374518, ranks A374638.
%Y A374699 - For weakly increasing leaders we have complement A374681.
%Y A374699 - For strictly increasing leaders we have complement complement A374679.
%Y A374699 - For strictly decreasing leaders we have complement A374680.
%Y A374699 A003242 counts anti-runs, ranks A333489.
%Y A374699 A106356 counts compositions by number of maximal anti-runs.
%Y A374699 A238279 counts compositions by number of maximal runs
%Y A374699 A238424 counts partitions whose first differences are an anti-run.
%Y A374699 A274174 counts contiguous compositions, ranks A374249.
%Y A374699 A333381 counts maximal anti-runs in standard compositions.
%Y A374699 Cf. A115029, A238343, A333213, A373949, A374515, A374632, A374635, A374700.
%K A374699 nonn,more
%O A374699 0,7
%A A374699 _Gus Wiseman_, Aug 06 2024