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 A374686 #11 Jul 28 2024 10:07:39
%S A374686 1,1,2,3,6,9,17,29,51,91,162,291,523,948,1712,3112,5656,10297,18763,
%T A374686 34217,62442,114006,208239,380465,695342,1271046,2323818,4249113,
%U A374686 7770389,14210991,25991853,47541734,86962675,159077005,291001483,532345978,973871397
%N A374686 Number of integer compositions of n whose leaders of strictly increasing runs are identical.
%C A374686 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 A374686 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 identical. For maxima instead of minima we have A374760. For all partitions (not just strict) we have A374704, for maxima A358905.
%H A374686 Andrew Howroyd, <a href="/A374686/b374686.txt">Table of n, a(n) for n = 0..1000</a>
%H A374686 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374686 The composition (2,3,2,2,3,4) has strictly increasing runs ((2,3),(2),(2,3,4)), with leaders (2,2,2), so is counted under a(16).
%e A374686 The a(0) = 1 through a(6) = 17 compositions:
%e A374686 () (1) (2) (3) (4) (5) (6)
%e A374686 (11) (12) (13) (14) (15)
%e A374686 (111) (22) (23) (24)
%e A374686 (112) (113) (33)
%e A374686 (121) (131) (114)
%e A374686 (1111) (1112) (123)
%e A374686 (1121) (141)
%e A374686 (1211) (222)
%e A374686 (11111) (1113)
%e A374686 (1131)
%e A374686 (1212)
%e A374686 (1311)
%e A374686 (11112)
%e A374686 (11121)
%e A374686 (11211)
%e A374686 (12111)
%e A374686 (111111)
%t A374686 Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n],SameQ@@First/@Split[#,Less]&]],{n,0,15}]
%o A374686 (PARI) seq(n) = Vec(1 + sum(k=1, n, 1/(1 - x^k*prod(j=k+1, n-k, 1 + x^j, 1 + O(x^(n-k+1))))-1)) \\ _Andrew Howroyd_, Jul 27 2024
%Y A374686 Ranked by A374685.
%Y A374686 Types of runs (instead of strictly increasing):
%Y A374686 - For leaders of identical runs we have A000005 for n > 0, ranks A272919.
%Y A374686 - For leaders of anti-runs we have A374517, ranks A374519.
%Y A374686 - For leaders of weakly increasing runs we have A374631, ranks A374633.
%Y A374686 - For leaders of weakly decreasing runs we have A374742, ranks A374744.
%Y A374686 - For leaders of strictly decreasing runs we have A374760, ranks A374759.
%Y A374686 Types of run-leaders (instead of identical):
%Y A374686 - For distinct leaders we have A374687, ranks A374698.
%Y A374686 - For strictly increasing leaders we have A374688.
%Y A374686 - For strictly decreasing leaders we have A374689.
%Y A374686 - For weakly increasing leaders we have A374690.
%Y A374686 - For weakly decreasing leaders we have A374697.
%Y A374686 A003242 counts anti-run compositions, ranks A333489.
%Y A374686 A011782 counts compositions.
%Y A374686 A238130, A238279, A333755 count compositions by number of runs.
%Y A374686 A274174 counts contiguous compositions, ranks A374249.
%Y A374686 A335456 counts patterns matched by compositions.
%Y A374686 A335548 counts non-contiguous compositions, ranks A374253.
%Y A374686 A373949 counts compositions by run-compressed sum, opposite A373951.
%Y A374686 A374683 lists leaders of strictly increasing runs of standard compositions.
%Y A374686 A374700 counts compositions by sum of leaders of strictly increasing runs.
%Y A374686 Cf. A000009, A106356, A188920, A189076, A238343, A304969, A333213, A374632, A374634, A374635, A374640.
%K A374686 nonn
%O A374686 0,3
%A A374686 _Gus Wiseman_, Jul 27 2024
%E A374686 a(26) onwards from _Andrew Howroyd_, Jul 27 2024