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Number of integer compositions of n whose leaders of anti-runs are weakly increasing.
11

%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