%I #5 Aug 02 2024 08:56:46
%S 1,1,2,4,7,14,27,50,96,185,353,672,1289,2466,4722,9052,17342,33244,
%T 63767,122325,234727,450553,864975,1660951,3190089,6128033
%N Number of integer compositions of n whose leaders of anti-runs are weakly increasing.
%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(5) = 14 compositions:
%e () (1) (2) (3) (4) (5)
%e (11) (12) (13) (14)
%e (21) (22) (23)
%e (111) (31) (32)
%e (112) (41)
%e (121) (113)
%e (1111) (122)
%e (131)
%e (212)
%e (221)
%e (1112)
%e (1121)
%e (1211)
%e (11111)
%t Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],LessEqual@@First/@Split[#,UnsameQ]&]],{n,0,15}]
%Y For partitions instead of compositions we have A034296.
%Y Other types of runs (instead of anti-):
%Y - For leaders of constant runs we have A000041.
%Y - For leaders of weakly decreasing runs we have A188900.
%Y - For leaders of weakly increasing runs we have A374635.
%Y - For leaders of strictly increasing runs we have A374690.
%Y - For leaders of strictly decreasing runs we have A374764.
%Y Other types of run-leaders (instead of weakly increasing):
%Y - For identical leaders we have A374517, ranks A374519.
%Y - For distinct leaders we have A374518, ranks A374638.
%Y - For strictly increasing leaders we have A374679.
%Y - For weakly decreasing leaders we have A374682.
%Y - For strictly decreasing leaders we have A374680.
%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 0,3
%A _Gus Wiseman_, Aug 01 2024