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 A374762 #12 Jul 31 2024 17:27:34
%S A374762 1,1,1,3,4,6,11,18,27,41,64,98,151,229,339,504,746,1097,1618,2372,
%T A374762 3451,5009,7233,10394,14905,21316,30396,43246,61369,86830,122529,
%U A374762 172457,242092,339062,473850,660829,919822,1277935,1772174,2453151,3389762,4675660,6438248
%N A374762 Number of integer compositions of n whose leaders of strictly decreasing runs are strictly increasing.
%C A374762 The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.
%C A374762 Also the number of ways to choose a strict integer partition of each part of an integer composition of n (A304969) such that the maxima are strictly decreasing. The weakly decreasing version is A374764.
%H A374762 Andrew Howroyd, <a href="/A374762/b374762.txt">Table of n, a(n) for n = 0..1000</a>
%H A374762 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A374762 G.f.: Product_{k>=1} (1 + x^k*Product_{j=1..k-1} (1 + x^j)). - _Andrew Howroyd_, Jul 31 2024
%e A374762 The a(0) = 1 through a(7) = 18 compositions:
%e A374762 () (1) (2) (3) (4) (5) (6) (7)
%e A374762 (12) (13) (14) (15) (16)
%e A374762 (21) (31) (23) (24) (25)
%e A374762 (121) (32) (42) (34)
%e A374762 (41) (51) (43)
%e A374762 (131) (123) (52)
%e A374762 (132) (61)
%e A374762 (141) (124)
%e A374762 (213) (142)
%e A374762 (231) (151)
%e A374762 (321) (214)
%e A374762 (232)
%e A374762 (241)
%e A374762 (421)
%e A374762 (1213)
%e A374762 (1231)
%e A374762 (1321)
%e A374762 (2131)
%t A374762 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Less@@First/@Split[#,Greater]&]],{n,0,15}]
%o A374762 (PARI) seq(n) = Vec(prod(k=1, n, 1 + x^k*prod(j=1, min(n-k,k-1), 1 + x^j, 1 + O(x^(n-k+1))))) \\ _Andrew Howroyd_, Jul 31 2024
%Y A374762 For partitions instead of compositions we have A000009.
%Y A374762 The weak version appears to be A188900.
%Y A374762 The opposite version is A374689.
%Y A374762 Other types of runs (instead of strictly decreasing):
%Y A374762 - For leaders of identical runs we have A000041.
%Y A374762 - For leaders of weakly increasing runs we have A374634.
%Y A374762 - For leaders of anti-runs we have A374679.
%Y A374762 Other types of run-leaders (instead of strictly increasing):
%Y A374762 - For identical leaders we have A374760, ranks A374759.
%Y A374762 - For distinct leaders we have A374761, ranks A374767.
%Y A374762 - For strictly decreasing leaders we have A374763.
%Y A374762 - For weakly increasing leaders we have A374764.
%Y A374762 - For weakly decreasing leaders we have A374765.
%Y A374762 A003242 counts anti-run compositions, ranks A333489.
%Y A374762 A011782 counts compositions.
%Y A374762 A238130, A238279, A333755 count compositions by number of runs.
%Y A374762 A274174 counts contiguous compositions, ranks A374249.
%Y A374762 A373949 counts compositions by run-compressed sum, opposite A373951.
%Y A374762 A374700 counts compositions by sum of leaders of strictly increasing runs.
%Y A374762 Cf. A106356, A188920, A189076, A238343, A261982, A333213, A374518, A374631, A374632, A374687, A374742, A374743.
%K A374762 nonn
%O A374762 0,4
%A A374762 _Gus Wiseman_, Jul 29 2024
%E A374762 a(24) onwards from _Andrew Howroyd_, Jul 31 2024