%I #5 Aug 06 2024 21:36:51
%S 0,0,0,0,1,3,7,18,43,96,211,463,992,2112,4462,9347,19495,40480,83690,
%T 172478,354455,726538,1486024,3033644,6182389,12580486
%N Number of integer compositions of n whose leaders of maximal anti-runs are not identical.
%C The leaders of maximal 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) = 0 through a(7) = 18 compositions:
%e . . . . (211) (122) (411) (133)
%e (311) (1122) (322)
%e (2111) (1221) (511)
%e (2112) (1222)
%e (2211) (2113)
%e (3111) (2311)
%e (21111) (3112)
%e (3211)
%e (4111)
%e (11122)
%e (11221)
%e (12211)
%e (21112)
%e (21121)
%e (21211)
%e (22111)
%e (31111)
%e (211111)
%t Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],!SameQ@@First/@Split[#,UnsameQ]&]],{n,0,15}]
%Y For partitions instead of compositions we have A239955.
%Y The complement is counted by A374517, ranks A374519.
%Y Compositions of this type are ranked by A374520, complement A374519.
%Y For distinct instead of identical leaders we have A374678, ranks A374639, complement A374518, ranks A374638.
%Y A003242 counts anti-runs, ranks A333489.
%Y A065120 gives leaders of standard compositions.
%Y A106356 counts compositions by number of maximal anti-runs.
%Y A238279 counts compositions by number of maximal runs
%Y A274174 counts contiguous compositions, ranks A374249.
%Y Cf. A034296, A188920, A189076, A238343, A238424, A272919, A333213, A373949, A374515, A374700.
%K nonn,more
%O 0,6
%A _Gus Wiseman_, Aug 06 2024