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Number of integer compositions of n whose leaders of maximal strictly increasing runs are not weakly decreasing.
5

%I #7 Aug 06 2024 21:37:18

%S 0,0,0,0,0,1,3,9,25,63,152,355,809,1804,3963,8590,18423,39161,82620,

%T 173198,361101,749326,1548609,3189132,6547190,13404613

%N Number of integer compositions of n whose leaders of maximal strictly increasing runs are not weakly decreasing.

%C The leaders of maximal strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing 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 composition y = (1,2,1,3,2,3) has strictly increasing runs ((1,2),(1,3),(2,3)), with leaders (1,1,2), which are not weakly decreasing, so y is counted under a(12).

%e The a(0) = 0 through a(8) = 25 compositions:

%e . . . . . (122) (132) (133) (143)

%e (1122) (142) (152)

%e (1221) (1132) (233)

%e (1222) (1133)

%e (1321) (1142)

%e (2122) (1223)

%e (11122) (1232)

%e (11221) (1322)

%e (12211) (1331)

%e (1421)

%e (2132)

%e (3122)

%e (11132)

%e (11222)

%e (11321)

%e (12122)

%e (12212)

%e (12221)

%e (13211)

%e (21122)

%e (21221)

%e (111122)

%e (111221)

%e (112211)

%e (122111)

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

%Y For leaders of constant runs we have A056823.

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

%Y The complement is counted by A374697.

%Y For leaders of anti-runs we have A374699, complement A374682.

%Y Other functional neighbors: A188920, A374764, A374765.

%Y A003242 counts anti-run compositions, ranks A333489.

%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 A374700 counts compositions by sum of leaders of strictly increasing runs.

%Y Cf. A106356, A238343, A261982, A333213, A374632, A374679, A374683, A374689.

%K nonn,more

%O 0,7

%A _Gus Wiseman_, Aug 06 2024