cp's OEIS Frontend

A374682 Number of integer compositions of n whose leaders of anti-runs are weakly decreasing.

%I A374682 #5 Aug 02 2024 08:56:28
%S A374682 1,1,2,4,8,15,30,59,114,222,434,844,1641,3189,6192,12020,23320,45213,
%T A374682 87624,169744,328684,636221,1231067,2381269,4604713,8901664
%N A374682 Number of integer compositions of n whose leaders of anti-runs are weakly decreasing.
%C A374682 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 A374682 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374682 The a(0) = 1 through a(5) = 15 compositions:
%e A374682   ()  (1)  (2)   (3)    (4)     (5)
%e A374682            (11)  (12)   (13)    (14)
%e A374682                  (21)   (22)    (23)
%e A374682                  (111)  (31)    (32)
%e A374682                         (112)   (41)
%e A374682                         (121)   (113)
%e A374682                         (211)   (131)
%e A374682                         (1111)  (212)
%e A374682                                 (221)
%e A374682                                 (311)
%e A374682                                 (1112)
%e A374682                                 (1121)
%e A374682                                 (1211)
%e A374682                                 (2111)
%e A374682                                 (11111)
%t A374682 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],GreaterEqual@@First/@Split[#,UnsameQ]&]],{n,0,15}]
%Y A374682 For reversed partitions instead of compositions we have A115029.
%Y A374682 The complement is A374699.
%Y A374682 Other types of runs (instead of anti-):
%Y A374682 - For leaders of identical runs we have A000041.
%Y A374682 - For leaders of weakly increasing runs we have A189076, complement A374636.
%Y A374682 - For leaders of weakly decreasing runs we have A374747.
%Y A374682 - For leaders of strictly decreasing runs we have A374765.
%Y A374682 - For leaders of strictly increasing runs we have A374697.
%Y A374682 Other types of run-leaders (instead of weakly decreasing):
%Y A374682 - For identical leaders we have A374517, ranks A374519.
%Y A374682 - For distinct leaders we have A374518, ranks A374638.
%Y A374682 - For weakly increasing leaders we have A374681.
%Y A374682 - For strictly increasing leaders we have A374679.
%Y A374682 - For strictly decreasing leaders we have A374680.
%Y A374682 A003242 counts anti-runs, ranks A333489.
%Y A374682 A106356 counts compositions by number of maximal anti-runs.
%Y A374682 A238279 counts compositions by number of maximal runs
%Y A374682 A238424 counts partitions whose first differences are an anti-run.
%Y A374682 A274174 counts contiguous compositions, ranks A374249.
%Y A374682 Cf. A238343, A333213, A333381, A373949, A374515, A374632, A374635, A374678, A374700, A374706.
%K A374682 nonn,more
%O A374682 0,3
%A A374682 _Gus Wiseman_, Aug 01 2024