cp's OEIS Frontend

A374680 Number of integer compositions of n whose leaders of anti-runs are strictly decreasing.

%I A374680 #5 Aug 02 2024 08:56:55
%S A374680 1,1,1,3,5,8,16,31,52,98,179,323,590,1078,1945,3531,6421,11621,21041,
%T A374680 38116,68904,124562,225138,406513,733710,1323803
%N A374680 Number of integer compositions of n whose leaders of anti-runs are strictly decreasing.
%C A374680 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 A374680 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374680 The a(0) = 1 through a(6) = 16 compositions:
%e A374680   ()  (1)  (2)  (3)   (4)    (5)    (6)
%e A374680                 (12)  (13)   (14)   (15)
%e A374680                 (21)  (31)   (23)   (24)
%e A374680                       (121)  (32)   (42)
%e A374680                       (211)  (41)   (51)
%e A374680                              (131)  (123)
%e A374680                              (212)  (132)
%e A374680                              (311)  (141)
%e A374680                                     (213)
%e A374680                                     (231)
%e A374680                                     (312)
%e A374680                                     (321)
%e A374680                                     (411)
%e A374680                                     (1212)
%e A374680                                     (2112)
%e A374680                                     (2121)
%t A374680 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Greater@@First/@Split[#,UnsameQ]&]],{n,0,15}]
%Y A374680 For distinct but not necessarily decreasing leaders we have A374518.
%Y A374680 For partitions instead of compositions we have A375133.
%Y A374680 Other types of runs (instead of anti-):
%Y A374680 - For leaders of identical runs we have A000041.
%Y A374680 - For leaders of weakly increasing runs we have A188920.
%Y A374680 - For leaders of weakly decreasing runs we have A374746.
%Y A374680 - For leaders of strictly decreasing runs we have A374763.
%Y A374680 - For leaders of strictly increasing runs we have A374689.
%Y A374680 Other types of run-leaders (instead of strictly decreasing):
%Y A374680 - For identical leaders we have A374517, ranks A374519.
%Y A374680 - For distinct leaders we have A374518, ranks A374638.
%Y A374680 - For weakly increasing leaders we have A374681.
%Y A374680 - For strictly increasing leaders we have A374679.
%Y A374680 - For weakly decreasing leaders we have A374682.
%Y A374680 A003242 counts anti-runs, ranks A333489.
%Y A374680 A106356 counts compositions by number of maximal anti-runs.
%Y A374680 A238279 counts compositions by number of maximal runs
%Y A374680 A238424 counts partitions whose first differences are an anti-run.
%Y A374680 A274174 counts contiguous compositions, ranks A374249.
%Y A374680 Cf. A189076, A238343, A333213, A333381, A373949, A374515, A374632, A374678, A374700, A374706.
%K A374680 nonn,more
%O A374680 0,4
%A A374680 _Gus Wiseman_, Aug 01 2024