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 A374763 #14 Dec 30 2024 21:53:21
%S A374763 1,1,1,2,3,4,6,10,15,22,32,47,71,106,156,227,328,473,683,986,1421,
%T A374763 2040,2916,4149,5882,8314,11727,16515,23221,32593,45655,63810,88979,
%U A374763 123789,171838,238055,329187,454451,626412,862164,1184917,1626124,2228324,3048982,4165640,5682847
%N A374763 Number of integer compositions of n whose leaders of strictly decreasing runs are themselves strictly decreasing.
%C A374763 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.
%H A374763 Andrew Howroyd, <a href="/A374763/b374763.txt">Table of n, a(n) for n = 0..1000</a>
%H A374763 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A374763 G.f.: Sum_{k>=0} x^k*Q(k,x) where Q(0,x) = 1 and Q(k,x) = Q(k-1,x) * (1 + x^k*Product_{j=1..k} (1 + x^j)) for k > 0. - _Andrew Howroyd_, Dec 30 2024
%e A374763 The composition (3,1,2,1,1) has strictly decreasing runs ((3,1),(2,1),(1)), with leaders (3,2,1), so is counted under a(8).
%e A374763 The a(0) = 1 through a(8) = 15 compositions:
%e A374763 () (1) (2) (3) (4) (5) (6) (7) (8)
%e A374763 (21) (31) (32) (42) (43) (53)
%e A374763 (211) (41) (51) (52) (62)
%e A374763 (311) (312) (61) (71)
%e A374763 (321) (322) (413)
%e A374763 (411) (412) (422)
%e A374763 (421) (431)
%e A374763 (511) (512)
%e A374763 (3121) (521)
%e A374763 (3211) (611)
%e A374763 (3212)
%e A374763 (3221)
%e A374763 (4121)
%e A374763 (4211)
%e A374763 (31211)
%t A374763 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Greater@@First/@Split[#,Greater]&]],{n,0,15}]
%o A374763 (PARI) seq(n)={ my(A=O(x*x^n), p=1+A, q=p, r=p); for(k=1, n\2, r += x^k*q; p *= 1 + x^k; q *= 1 + x^k*p); Vec(r + x^(n\2+1)*q/(1-x)) } \\ _Andrew Howroyd_, Dec 30 2024
%Y A374763 The opposite version is A374688.
%Y A374763 The weak version is A374747.
%Y A374763 For partitions instead of compositions we have A375133.
%Y A374763 Other types of runs (instead of strictly decreasing):
%Y A374763 - For leaders of identical runs we have A000041.
%Y A374763 - For leaders of weakly increasing runs we appear to have A188920.
%Y A374763 - For leaders of anti-runs we have A374680.
%Y A374763 - For leaders of strictly increasing runs we have A374689.
%Y A374763 - For leaders of weakly decreasing runs we have A374746.
%Y A374763 Other types of run-leaders (instead of strictly decreasing):
%Y A374763 - For identical leaders we have A374760, ranks A374759.
%Y A374763 - For distinct leaders we have A374761, ranks A374767.
%Y A374763 - For strictly increasing leaders we have A374762.
%Y A374763 - For weakly increasing leaders we have A374764.
%Y A374763 - For weakly decreasing leaders we have A374765.
%Y A374763 A003242 counts anti-run compositions, ranks A333489.
%Y A374763 A011782 counts compositions.
%Y A374763 A238130, A238279, A333755 count compositions by number of runs.
%Y A374763 A274174 counts contiguous compositions, ranks A374249.
%Y A374763 A373949 counts compositions by run-compressed sum, opposite A373951.
%Y A374763 Cf. A000009, A106356, A188900, A238343, A261982, A333213, A374518, A374632, A374635, A374687, A374742, A374743.
%K A374763 nonn
%O A374763 0,4
%A A374763 _Gus Wiseman_, Jul 30 2024
%E A374763 a(24) onwards from _Andrew Howroyd_, Dec 30 2024