This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A374689 #19 Jul 31 2024 22:11:41
%S A374689 1,1,1,3,3,6,10,13,21,32,48,66,101,144,207,298,415,592,833,1163,1615,
%T A374689 2247,3088,4259,5845,7977,10862,14752,19969,26941,36310,48725,65279,
%U A374689 87228,116274,154660,205305,271879,359400,474157,624257,820450,1076357,1409598
%N A374689 Number of integer compositions of n whose leaders of strictly increasing runs are strictly decreasing.
%C A374689 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 A374689 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. The weakly decreasing version is A374697.
%H A374689 Andrew Howroyd, <a href="/A374689/b374689.txt">Table of n, a(n) for n = 0..1000</a>
%H A374689 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A374689 G.f.: Product_{i>0} (1 + (x^i)*Product_{j>i} (1 + x^j)). - _John Tyler Rascoe_, Jul 29 2024
%e A374689 The a(0) = 1 through a(8) = 21 compositions:
%e A374689 () (1) (2) (3) (4) (5) (6) (7) (8)
%e A374689 (12) (13) (14) (15) (16) (17)
%e A374689 (21) (31) (23) (24) (25) (26)
%e A374689 (32) (42) (34) (35)
%e A374689 (41) (51) (43) (53)
%e A374689 (212) (123) (52) (62)
%e A374689 (213) (61) (71)
%e A374689 (231) (124) (125)
%e A374689 (312) (214) (134)
%e A374689 (321) (241) (215)
%e A374689 (313) (251)
%e A374689 (412) (314)
%e A374689 (421) (323)
%e A374689 (341)
%e A374689 (413)
%e A374689 (431)
%e A374689 (512)
%e A374689 (521)
%e A374689 (2123)
%e A374689 (2312)
%e A374689 (3212)
%t A374689 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Greater@@First/@Split[#,Less]&]],{n,0,15}]
%o A374689 (PARI)
%o A374689 C_x(N) = {my(x='x+O('x^N), h=prod(i=1,N, 1+(x^i)*prod(j=i+1,N, 1+x^j))); Vec(h)}
%o A374689 C_x(50) \\ _John Tyler Rascoe_, Jul 29 2024
%Y A374689 The weak version appears to be A189076.
%Y A374689 Ranked by positions of strictly decreasing rows in A374683.
%Y A374689 The opposite version is A374762.
%Y A374689 Types of runs (instead of strictly increasing):
%Y A374689 - For leaders of identical runs we have A000041.
%Y A374689 - For leaders of anti-runs we have A374680.
%Y A374689 - For leaders of weakly increasing runs we have A188920.
%Y A374689 - For leaders of weakly decreasing runs we have A374746.
%Y A374689 - For leaders of strictly decreasing runs we have A374763.
%Y A374689 Types of run-leaders (instead of strictly decreasing):
%Y A374689 - For identical leaders we have A374686, ranks A374685.
%Y A374689 - For distinct leaders we have A374687, ranks A374698.
%Y A374689 - For strictly increasing leaders we have A374688.
%Y A374689 - For weakly increasing leaders we have A374690.
%Y A374689 - For weakly decreasing leaders we have A374697.
%Y A374689 A003242 counts anti-run compositions, ranks A333489.
%Y A374689 A011782 counts compositions.
%Y A374689 A238130, A238279, A333755 count compositions by number of runs.
%Y A374689 A335456 counts patterns matched by compositions.
%Y A374689 A373949 counts compositions by run-compressed sum, opposite A373951.
%Y A374689 A374700 counts compositions by sum of leaders of strictly increasing runs.
%Y A374689 Cf. A000009, A106356, A238343, A261982, A304969, A333213, A374632, A374635, A374640, A374679.
%K A374689 nonn
%O A374689 0,4
%A A374689 _Gus Wiseman_, Jul 27 2024
%E A374689 a(26) onwards from _John Tyler Rascoe_, Jul 29 2024