A374682 - OEIS

%I #5 Aug 02 2024 08:56:28

%S 1,1,2,4,8,15,30,59,114,222,434,844,1641,3189,6192,12020,23320,45213,

%T 87624,169744,328684,636221,1231067,2381269,4604713,8901664

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

%C 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 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.

%e The a(0) = 1 through a(5) = 15 compositions:

%e   ()  (1)  (2)   (3)    (4)     (5)

%e            (11)  (12)   (13)    (14)

%e                  (21)   (22)    (23)

%e                  (111)  (31)    (32)

%e                         (112)   (41)

%e                         (121)   (113)

%e                         (211)   (131)

%e                         (1111)  (212)

%e                                 (221)

%e                                 (311)

%e                                 (1112)

%e                                 (1121)

%e                                 (1211)

%e                                 (2111)

%e                                 (11111)

%t Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],GreaterEqual@@First/@Split[#,UnsameQ]&]],{n,0,15}]

%Y For reversed partitions instead of compositions we have A115029.

%Y The complement is A374699.

%Y Other types of runs (instead of anti-):

%Y - For leaders of identical runs we have A000041.

%Y - For leaders of weakly increasing runs we have A189076, complement A374636.

%Y - For leaders of weakly decreasing runs we have A374747.

%Y - For leaders of strictly decreasing runs we have A374765.

%Y - For leaders of strictly increasing runs we have A374697.

%Y Other types of run-leaders (instead of weakly decreasing):

%Y - For identical leaders we have A374517, ranks A374519.

%Y - For distinct leaders we have A374518, ranks A374638.

%Y - For weakly increasing leaders we have A374681.

%Y - For strictly increasing leaders we have A374679.

%Y - For strictly decreasing leaders we have A374680.

%Y A003242 counts anti-runs, ranks A333489.

%Y A106356 counts compositions by number of maximal anti-runs.

%Y A238279 counts compositions by number of maximal runs

%Y A238424 counts partitions whose first differences are an anti-run.

%Y A274174 counts contiguous compositions, ranks A374249.

%Y Cf. A238343, A333213, A333381, A373949, A374515, A374632, A374635, A374678, A374700, A374706.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Aug 01 2024