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Number of integer compositions of n whose leaders of maximal anti-runs are not identical.
10

%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