cp's OEIS Frontend

A374518 Number of integer compositions of n whose leaders of anti-runs are distinct.

%I A374518 #7 Aug 02 2024 08:58:34
%S A374518 1,1,1,3,5,9,17,32,58,112,201,371,694,1276,2342,4330,7958,14613,26866,
%T A374518 49303,90369,165646,303342,555056,1015069,1855230
%N A374518 Number of integer compositions of n whose leaders of anti-runs are distinct.
%C A374518 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 A374518 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374518 The a(0) = 1 through a(6) = 17 compositions:
%e A374518   ()  (1)  (2)  (3)   (4)    (5)    (6)
%e A374518                 (12)  (13)   (14)   (15)
%e A374518                 (21)  (31)   (23)   (24)
%e A374518                       (121)  (32)   (42)
%e A374518                       (211)  (41)   (51)
%e A374518                              (122)  (123)
%e A374518                              (131)  (132)
%e A374518                              (212)  (141)
%e A374518                              (311)  (213)
%e A374518                                     (231)
%e A374518                                     (312)
%e A374518                                     (321)
%e A374518                                     (411)
%e A374518                                     (1212)
%e A374518                                     (1221)
%e A374518                                     (2112)
%e A374518                                     (2121)
%t A374518 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],UnsameQ@@First/@Split[#,UnsameQ]&]],{n,0,15}]
%Y A374518 These compositions have ranks A374638.
%Y A374518 The complement is counted by A374678.
%Y A374518 For partitions instead of compositions we have A375133.
%Y A374518 Other types of runs (instead of anti-):
%Y A374518 - For leaders of identical runs we have A274174, ranks A374249.
%Y A374518 - For leaders of weakly increasing runs we have A374632, ranks A374768.
%Y A374518 - For leaders of strictly increasing runs we have A374687, ranks A374698.
%Y A374518 - For leaders of weakly decreasing runs we have A374743, ranks A374701.
%Y A374518 - For leaders of strictly decreasing runs we have A374761, ranks A374767.
%Y A374518 Other types of run-leaders (instead of distinct):
%Y A374518 - For identical leaders we have A374517.
%Y A374518 - For weakly increasing leaders we have A374681.
%Y A374518 - For strictly increasing leaders we have A374679.
%Y A374518 - For weakly decreasing leaders we have A374682.
%Y A374518 - For strictly decreasing leaders we have A374680.
%Y A374518 A003242 counts anti-runs, ranks A333489.
%Y A374518 A106356 counts compositions by number of maximal anti-runs.
%Y A374518 A238279 counts compositions by number of maximal runs
%Y A374518 A238424 counts partitions whose first differences are an anti-run.
%Y A374518 Cf. A188920, A233564, A238343, A333213, A333381, A373949, A374515.
%K A374518 nonn
%O A374518 0,4
%A A374518 _Gus Wiseman_, Aug 01 2024