cp's OEIS Frontend

A374681 Number of integer compositions of n whose leaders of anti-runs are weakly increasing.

%I A374681 #5 Aug 02 2024 08:56:46
%S A374681 1,1,2,4,7,14,27,50,96,185,353,672,1289,2466,4722,9052,17342,33244,
%T A374681 63767,122325,234727,450553,864975,1660951,3190089,6128033
%N A374681 Number of integer compositions of n whose leaders of anti-runs are weakly increasing.
%C A374681 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 A374681 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374681 The a(0) = 1 through a(5) = 14 compositions:
%e A374681   ()  (1)  (2)   (3)    (4)     (5)
%e A374681            (11)  (12)   (13)    (14)
%e A374681                  (21)   (22)    (23)
%e A374681                  (111)  (31)    (32)
%e A374681                         (112)   (41)
%e A374681                         (121)   (113)
%e A374681                         (1111)  (122)
%e A374681                                 (131)
%e A374681                                 (212)
%e A374681                                 (221)
%e A374681                                 (1112)
%e A374681                                 (1121)
%e A374681                                 (1211)
%e A374681                                 (11111)
%t A374681 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],LessEqual@@First/@Split[#,UnsameQ]&]],{n,0,15}]
%Y A374681 For partitions instead of compositions we have A034296.
%Y A374681 Other types of runs (instead of anti-):
%Y A374681 - For leaders of constant runs we have A000041.
%Y A374681 - For leaders of weakly decreasing runs we have A188900.
%Y A374681 - For leaders of weakly increasing runs we have A374635.
%Y A374681 - For leaders of strictly increasing runs we have A374690.
%Y A374681 - For leaders of strictly decreasing runs we have A374764.
%Y A374681 Other types of run-leaders (instead of weakly increasing):
%Y A374681 - For identical leaders we have A374517, ranks A374519.
%Y A374681 - For distinct leaders we have A374518, ranks A374638.
%Y A374681 - For strictly increasing leaders we have A374679.
%Y A374681 - For weakly decreasing leaders we have A374682.
%Y A374681 - For strictly decreasing leaders we have A374680.
%Y A374681 A003242 counts anti-runs, ranks A333489.
%Y A374681 A106356 counts compositions by number of maximal anti-runs.
%Y A374681 A238279 counts compositions by number of maximal runs
%Y A374681 A238424 counts partitions whose first differences are an anti-run.
%Y A374681 A274174 counts contiguous compositions, ranks A374249.
%Y A374681 Cf. A189076, A238343, A333213, A333381, A373949, A374515, A374632, A374678, A374700, A374706.
%K A374681 nonn,more
%O A374681 0,3
%A A374681 _Gus Wiseman_, Aug 01 2024