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A374640 Number of integer compositions of n whose leaders of maximal anti-runs are not identical.

%I A374640 #5 Aug 06 2024 21:36:51
%S A374640 0,0,0,0,1,3,7,18,43,96,211,463,992,2112,4462,9347,19495,40480,83690,
%T A374640 172478,354455,726538,1486024,3033644,6182389,12580486
%N A374640 Number of integer compositions of n whose leaders of maximal anti-runs are not identical.
%C A374640 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 A374640 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374640 The a(0) = 0 through a(7) = 18 compositions:
%e A374640   .  .  .  .  (211)  (122)   (411)    (133)
%e A374640                      (311)   (1122)   (322)
%e A374640                      (2111)  (1221)   (511)
%e A374640                              (2112)   (1222)
%e A374640                              (2211)   (2113)
%e A374640                              (3111)   (2311)
%e A374640                              (21111)  (3112)
%e A374640                                       (3211)
%e A374640                                       (4111)
%e A374640                                       (11122)
%e A374640                                       (11221)
%e A374640                                       (12211)
%e A374640                                       (21112)
%e A374640                                       (21121)
%e A374640                                       (21211)
%e A374640                                       (22111)
%e A374640                                       (31111)
%e A374640                                       (211111)
%t A374640 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],!SameQ@@First/@Split[#,UnsameQ]&]],{n,0,15}]
%Y A374640 For partitions instead of compositions we have A239955.
%Y A374640 The complement is counted by A374517, ranks A374519.
%Y A374640 Compositions of this type are ranked by A374520, complement A374519.
%Y A374640 For distinct instead of identical leaders we have A374678, ranks A374639, complement A374518, ranks A374638.
%Y A374640 A003242 counts anti-runs, ranks A333489.
%Y A374640 A065120 gives leaders of standard compositions.
%Y A374640 A106356 counts compositions by number of maximal anti-runs.
%Y A374640 A238279 counts compositions by number of maximal runs
%Y A374640 A274174 counts contiguous compositions, ranks A374249.
%Y A374640 Cf. A034296, A188920, A189076, A238343, A238424, A272919, A333213, A373949, A374515, A374700.
%K A374640 nonn,more
%O A374640 0,6
%A A374640 _Gus Wiseman_, Aug 06 2024