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 A374764 #14 Jul 31 2024 17:27:38
%S A374764 1,1,2,4,7,13,23,40,69,118,199,333,553,911,1492,2428,3928,6323,10129,
%T A374764 16151,25646,40560,63905,100332,156995,244877,380803,590479,913100,
%U A374764 1408309,2166671,3325445,5092283,7780751,11863546,18052080,27415291,41556849,62879053,94975305,143213145
%N A374764 Number of integer compositions of n whose leaders of strictly decreasing runs are weakly increasing.
%C A374764 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 A374764 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 weakly increasing [but weakly decreasing works too]. The strictly increasing version is A374762.
%H A374764 Andrew Howroyd, <a href="/A374764/b374764.txt">Table of n, a(n) for n = 0..1000</a>
%H A374764 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A374764 G.f.: 1/(Product_{k>=1} (1 - x^k*Product_{j=1..k-1} (1 + x^j))). - _Andrew Howroyd_, Jul 31 2024
%e A374764 The composition (1,1,2,1) has strictly decreasing runs ((1),(1),(2,1)) with leaders (1,1,2) so is counted under a(5).
%e A374764 The composition (1,2,1,1) has strictly decreasing runs ((1),(2,1),(1)) with leaders (1,2,1) so is not counted under a(5).
%e A374764 The a(0) = 1 through a(5) = 13 compositions:
%e A374764 () (1) (2) (3) (4) (5)
%e A374764 (11) (12) (13) (14)
%e A374764 (21) (22) (23)
%e A374764 (111) (31) (32)
%e A374764 (112) (41)
%e A374764 (121) (113)
%e A374764 (1111) (122)
%e A374764 (131)
%e A374764 (212)
%e A374764 (221)
%e A374764 (1112)
%e A374764 (1121)
%e A374764 (11111)
%t A374764 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],LessEqual@@First/@Split[#,Greater]&]],{n,0,15}]
%o A374764 (PARI) seq(n) = Vec(1/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 A374764 For partitions instead of compositions we have A034296.
%Y A374764 For strictly increasing leaders we have A374688.
%Y A374764 The opposite version is A374697.
%Y A374764 Other types of runs (instead of strictly decreasing):
%Y A374764 - For leaders of identical runs we have A000041.
%Y A374764 - For leaders of anti-runs we have A374681.
%Y A374764 - For leaders of weakly increasing runs we have A374635.
%Y A374764 - For leaders of strictly increasing runs we have A374690.
%Y A374764 - For leaders of weakly decreasing runs we have A188900.
%Y A374764 Other types of run-leaders (instead of weakly increasing):
%Y A374764 - For identical leaders we have A374760, ranks A374759.
%Y A374764 - For distinct leaders we have A374761, ranks A374767.
%Y A374764 - For strictly increasing leaders we have A374762.
%Y A374764 - For weakly decreasing leaders we have A374765.
%Y A374764 - For strictly decreasing leaders we have A374763.
%Y A374764 A003242 counts anti-run compositions, ranks A333489.
%Y A374764 A011782 counts compositions.
%Y A374764 A238130, A238279, A333755 count compositions by number of runs.
%Y A374764 A274174 counts contiguous compositions, ranks A374249.
%Y A374764 A335548 counts non-contiguous compositions, ranks A374253.
%Y A374764 A373949 counts compositions by run-compressed sum, opposite A373951.
%Y A374764 Cf. A106356, A188920, A238343, A261982, A333213, A374687, A374679, A374680, A374742, A374743, A374747.
%K A374764 nonn
%O A374764 0,3
%A A374764 _Gus Wiseman_, Jul 30 2024
%E A374764 a(24) onwards from _Andrew Howroyd_, Jul 31 2024