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Number of integer compositions of n whose leaders of anti-runs are distinct.
34

%I #7 Aug 02 2024 08:58:34

%S 1,1,1,3,5,9,17,32,58,112,201,371,694,1276,2342,4330,7958,14613,26866,

%T 49303,90369,165646,303342,555056,1015069,1855230

%N Number of integer compositions of n whose leaders of anti-runs are distinct.

%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) = 17 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 (122) (123)

%e (131) (132)

%e (212) (141)

%e (311) (213)

%e (231)

%e (312)

%e (321)

%e (411)

%e (1212)

%e (1221)

%e (2112)

%e (2121)

%t Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],UnsameQ@@First/@Split[#,UnsameQ]&]],{n,0,15}]

%Y These compositions have ranks A374638.

%Y The complement is counted by A374678.

%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 A274174, ranks A374249.

%Y - For leaders of weakly increasing runs we have A374632, ranks A374768.

%Y - For leaders of strictly increasing runs we have A374687, ranks A374698.

%Y - For leaders of weakly decreasing runs we have A374743, ranks A374701.

%Y - For leaders of strictly decreasing runs we have A374761, ranks A374767.

%Y Other types of run-leaders (instead of distinct):

%Y - For identical leaders we have A374517.

%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 - 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 Cf. A188920, A233564, A238343, A333213, A333381, A373949, A374515.

%K nonn

%O 0,4

%A _Gus Wiseman_, Aug 01 2024