%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