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%I #5 Aug 02 2024 08:56:55
%S 1,1,1,3,5,8,16,31,52,98,179,323,590,1078,1945,3531,6421,11621,21041,
%T 38116,68904,124562,225138,406513,733710,1323803
%N Number of integer compositions of n whose leaders of anti-runs are strictly decreasing.
%C The leaders of 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 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e The a(0) = 1 through a(6) = 16 compositions:
%e () (1) (2) (3) (4) (5) (6)
%e (12) (13) (14) (15)
%e (21) (31) (23) (24)
%e (121) (32) (42)
%e (211) (41) (51)
%e (131) (123)
%e (212) (132)
%e (311) (141)
%e (213)
%e (231)
%e (312)
%e (321)
%e (411)
%e (1212)
%e (2112)
%e (2121)
%t Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Greater@@First/@Split[#,UnsameQ]&]],{n,0,15}]
%Y For distinct but not necessarily decreasing leaders we have A374518.
%Y For partitions instead of compositions we have A375133.
%Y Other types of runs (instead of anti-):
%Y - For leaders of identical runs we have A000041.
%Y - For leaders of weakly increasing runs we have A188920.
%Y - For leaders of weakly decreasing runs we have A374746.
%Y - For leaders of strictly decreasing runs we have A374763.
%Y - For leaders of strictly increasing runs we have A374689.
%Y Other types of run-leaders (instead of strictly decreasing):
%Y - For identical leaders we have A374517, ranks A374519.
%Y - For distinct leaders we have A374518, ranks A374638.
%Y - For weakly increasing leaders we have A374681.
%Y - For strictly increasing leaders we have A374679.
%Y - For weakly decreasing leaders we have A374682.
%Y A003242 counts anti-runs, ranks A333489.
%Y A106356 counts compositions by number of maximal anti-runs.
%Y A238279 counts compositions by number of maximal runs
%Y A238424 counts partitions whose first differences are an anti-run.
%Y A274174 counts contiguous compositions, ranks A374249.
%Y Cf. A189076, A238343, A333213, A333381, A373949, A374515, A374632, A374678, A374700, A374706.
%K nonn,more
%O 0,4
%A _Gus Wiseman_, Aug 01 2024
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