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%I #11 Aug 23 2024 08:40:32
%S 0,0,0,0,0,0,1,4,14,41,110,278,673,1576,3599,8055,17732,38509,82683,
%T 175830,370856,776723,1616945,3348500,6902905,14174198,29004911,
%U 59175625,120414435,244468774,495340191,1001911626,2023473267,4081241473,8222198324,16548146045,33276169507
%N Number of integer compositions of n that match the dashed pattern 3-12.
%C First differs from the non-dashed version A335514 at a(9) = 41, A335514(9) = 42, due to the composition (3,1,3,2).
%C Also the number of integer compositions of n whose leaders of weakly decreasing runs are not weakly increasing. For example, the composition q = (1,1,2,1,2,2,1,3) has maximal weakly decreasing runs ((1,1),(2,1),(2,2,1),(3)), with leaders (1,2,2,3), which are weakly increasing, so q is not counted under a(13); also q does not match 3-12. On the other hand, the reverse is (3,1,2,2,1,2,1,1), with maximal weakly decreasing runs ((3,1),(2,2,1),(2,1,1)), with leaders (3,2,2), which are not weakly increasing, so it is counted under a(13); meanwhile it matches 3-12, as required.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>.
%H Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F a(n>0) = 2^(n-1) - A188900(n).
%e The a(0) = 0 through a(8) = 14 compositions:
%e . . . . . . (312) (412) (413)
%e (1312) (512)
%e (3112) (1412)
%e (3121) (2312)
%e (3122)
%e (3212)
%e (4112)
%e (4121)
%e (11312)
%e (13112)
%e (13121)
%e (31112)
%e (31121)
%e (31211)
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !LessEqual@@First/@Split[#,GreaterEqual]&]],{n,0,15}]
%t - or -
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], MatchQ[#,{___,z_,___,x_,y_,___}/;x<y<z]&]],{n,0,15}] (*3-12*)
%Y For leaders of identical runs we have A056823.
%Y The complement is counted by A188900.
%Y The non-dashed version is A335514, ranks A335479.
%Y Ranks are positions of non-weakly increasing rows in A374740.
%Y A003242 counts anti-run compositions, ranks A333489.
%Y A011782 counts compositions.
%Y Counting compositions by number of runs: A238130, A238279, A333755.
%Y A373949 counts compositions by run-compressed sum, opposite A373951.
%Y Cf. A106356, A188920, A189076, A189077, A238343, A333213, A335548, A374629, A374637, A374679, A374748.
%K nonn
%O 0,8
%A _Gus Wiseman_, Aug 22 2024