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%I #14 Aug 08 2024 17:42:27
%S 1,1,1,2,2,4,5,7,11,16,21,31,45,63,87,122,170,238,328,449,616,844,
%T 1151,1565,2121,2861,3855,5183,6953,9299,12407,16513,21935,29078,
%U 38468,50793,66935,88037,115577,151473,198175,258852,337560,439507,571355,741631
%N Number of integer compositions of n whose leaders of strictly increasing runs are themselves strictly increasing.
%C The leaders of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.
%C Also the number of ways to choose a strict integer partition of each part of an integer composition of n (A304969) such that the minima are strictly decreasing.
%H Christian Sievers, <a href="/A374688/b374688.txt">Table of n, a(n) for n = 0..500</a>
%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(9) = 16 compositions:
%e () (1) (2) (3) (4) (5) (6) (7) (8) (9)
%e (12) (13) (14) (15) (16) (17) (18)
%e (23) (24) (25) (26) (27)
%e (122) (123) (34) (35) (36)
%e (132) (124) (125) (45)
%e (133) (134) (126)
%e (142) (143) (135)
%e (152) (144)
%e (233) (153)
%e (1223) (162)
%e (1232) (234)
%e (243)
%e (1224)
%e (1233)
%e (1242)
%e (1323)
%t Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Less@@First/@Split[#,Less]&]],{n,0,15}]
%Y The weak version is A374635.
%Y Ranked by positions of strictly increasing rows in A374683 (sums A374684).
%Y The opposite version is A374763.
%Y Types of runs (instead of strictly increasing):
%Y - For leaders of identical runs we have A000041.
%Y - For leaders of anti-runs we have A374679.
%Y - For leaders of weakly increasing runs we have A374634.
%Y - For leaders of weakly decreasing runs we have A374745.
%Y - For leaders of strictly decreasing runs we have A374762.
%Y Types of run-leaders (instead of strictly increasing):
%Y - For identical leaders we have A374686, ranks A374685.
%Y - For distinct leaders we have A374687, ranks A374698.
%Y - For strictly decreasing leaders we have A374689.
%Y - For weakly increasing leaders we have A374690.
%Y - For weakly decreasing leaders we have A374697.
%Y A003242 counts anti-run compositions, ranks A333489.
%Y A011782 counts compositions.
%Y A238130, A238279, A333755 count compositions by number of runs.
%Y A373949 counts compositions by run-compressed sum, opposite A373951.
%Y A374700 counts compositions by sum of leaders of strictly increasing runs.
%Y Cf. A000009, A106356, A188920, A189076, A238343, A261982, A333213, A374632.
%K nonn,changed
%O 0,4
%A _Gus Wiseman_, Jul 27 2024
%E a(26) and beyond from _Christian Sievers_, Aug 08 2024
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