cp's OEIS Frontend

A374678 Number of integer compositions of n whose leaders of maximal anti-runs are not distinct.

%I A374678 #6 Aug 06 2024 21:36:58
%S A374678 0,0,1,1,3,7,15,32,70,144,311,653,1354,2820,5850,12054,24810,50923,
%T A374678 104206,212841,433919,882930,1793810,3639248,7373539,14921986
%N A374678 Number of integer compositions of n whose leaders of maximal anti-runs are not distinct.
%C A374678 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 A374678 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374678 The anti-runs of y = (1,1,2,2) are ((1),(1,2),(2)) with leaders (1,1,2) so y is counted under a(6).
%e A374678 The a(0) = 0 through a(6) = 15 compositions:
%e A374678   .  .  (11)  (111)  (22)    (113)    (33)
%e A374678                      (112)   (221)    (114)
%e A374678                      (1111)  (1112)   (222)
%e A374678                              (1121)   (1113)
%e A374678                              (1211)   (1122)
%e A374678                              (2111)   (1131)
%e A374678                              (11111)  (1311)
%e A374678                                       (2211)
%e A374678                                       (3111)
%e A374678                                       (11112)
%e A374678                                       (11121)
%e A374678                                       (11211)
%e A374678                                       (12111)
%e A374678                                       (21111)
%e A374678                                       (111111)
%t A374678 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],!UnsameQ@@First/@Split[#,UnsameQ]&]],{n,0,15}]
%Y A374678 For constant runs we have A335548, complement A274174, ranks A374249.
%Y A374678 The complement is counted by A374518, ranks A374638.
%Y A374678 For weakly increasing runs we have complement A374632, ranks A374768.
%Y A374678 Compositions of this type are ranked by A374639.
%Y A374678 For identical instead of distinct leaders we have A374640, ranks A374520, complement A374517, ranks A374519.
%Y A374678 A003242 counts anti-runs, ranks A333489.
%Y A374678 A065120 gives leaders of standard compositions.
%Y A374678 A106356 counts compositions by number of maximal anti-runs.
%Y A374678 A238279 counts compositions by number of maximal runs
%Y A374678 A274174 counts contiguous compositions, ranks A374249.
%Y A374678 Cf. A034296, A188920, A189076, A238343, A239955, A333213, A373949, A374515.
%K A374678 nonn,more
%O A374678 0,5
%A A374678 _Gus Wiseman_, Aug 06 2024