|
%I #8 Jul 26 2024 08:46:17
%S 1,1,2,3,5,7,12,18,31,51,86,143,241,397,657,1082,1771,2889,4697,7605,
%T 12269,19720,31580,50412,80205,127208,201149,317171,498717,782076,
%U 1223230,1908381,2969950,4610949,7141972,11037276,17019617,26188490,40213388,61624824
%N Number of integer compositions of n whose leaders of weakly decreasing runs are strictly decreasing.
%C The weakly decreasing run-leaders of a sequence are obtained by splitting it into maximal weakly decreasing subsequences 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(7) = 18 compositions:
%e () (1) (2) (3) (4) (5) (6) (7)
%e (11) (21) (22) (32) (33) (43)
%e (111) (31) (41) (42) (52)
%e (211) (221) (51) (61)
%e (1111) (311) (222) (322)
%e (2111) (312) (331)
%e (11111) (321) (412)
%e (411) (421)
%e (2211) (511)
%e (3111) (2221)
%e (21111) (3112)
%e (111111) (3121)
%e (3211)
%e (4111)
%e (22111)
%e (31111)
%e (211111)
%e (1111111)
%t Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Greater@@First/@Split[#,GreaterEqual]&]],{n,0,15}]
%Y Ranked by positions of strictly decreasing rows in A374740, opp. A374629.
%Y Types of runs (instead of weakly decreasing):
%Y - For leaders of identical runs we have A000041.
%Y - For leaders of weakly increasing runs we have A188920.
%Y - For leaders of anti-runs we have A374680.
%Y - For leaders of strictly increasing runs we have A374689.
%Y - For leaders of strictly decreasing runs we have A374763.
%Y Types of run-leaders (instead of strictly decreasing):
%Y - For weakly increasing leaders we appear to have A188900.
%Y - For identical leaders we have A374742.
%Y - For distinct leaders we have A374743, ranks A374701.
%Y - For strictly increasing leaders we have A374745, opposite A374634.
%Y - For weakly decreasing leaders we have A374747.
%Y A011782 counts compositions.
%Y A238130, A238279, A333755 count compositions by number of runs.
%Y A335456 counts patterns matched by compositions.
%Y A373949 counts compositions by run-compressed sum, opposite A373951.
%Y A374748 counts compositions by sum of leaders of weakly decreasing runs.
%Y Cf. A000009, A003242, A106356, A189076, A238343, A261982, A333213, A358836, A374632, A374635, A374741.
%K nonn,new
%O 0,3
%A _Gus Wiseman_, Jul 26 2024
%E a(24)-a(39) from _Alois P. Heinz_, Jul 26 2024
| 