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 A351013 #10 Feb 16 2022 15:39:32
%S A351013 1,1,2,4,7,14,26,48,88,161,294,512,970,1634,2954,5156,9119,15618,
%T A351013 27354,46674,80130,138078,232286,394966,665552,1123231,1869714,
%U A351013 3146410,5186556,8620936,14324366,23529274,38564554,63246744,103578914,167860584,274465845
%N A351013 Number of integer compositions of n with all distinct runs.
%H A351013 Andrew Howroyd, <a href="/A351013/b351013.txt">Table of n, a(n) for n = 0..1000</a>
%H A351013 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>
%e A351013 The a(1) = 1 through a(5) = 14 compositions:
%e A351013 (1) (2) (3) (4) (5)
%e A351013 (1,1) (1,2) (1,3) (1,4)
%e A351013 (2,1) (2,2) (2,3)
%e A351013 (1,1,1) (3,1) (3,2)
%e A351013 (1,1,2) (4,1)
%e A351013 (2,1,1) (1,1,3)
%e A351013 (1,1,1,1) (1,2,2)
%e A351013 (2,2,1)
%e A351013 (3,1,1)
%e A351013 (1,1,1,2)
%e A351013 (1,1,2,1)
%e A351013 (1,2,1,1)
%e A351013 (2,1,1,1)
%e A351013 (1,1,1,1,1)
%e A351013 For example, the composition c = (3,1,1,1,1,2,1,1,3,4,1,1) has runs (3), (1,1,1,1), (2), (1,1), (3), (4), (1,1), and since (3) and (1,1) both appear twice, c is not counted under a(20).
%t A351013 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Split[#]&]],{n,0,10}]
%o A351013 (PARI) \\ here LahI is A111596 as row polynomials.
%o A351013 LahI(n,y) = {sum(k=1, n, y^k*(-1)^(n-k)*(n!/k!)*binomial(n-1, k-1))}
%o A351013 S(n) = {my(p=prod(k=1, n, 1 + y*x^k + O(x*x^n))); 1 + sum(i=1, (sqrtint(8*n+1)-1)\2, polcoef(p,i,y)*LahI(i,y))}
%o A351013 seq(n)={my(q=S(n)); [subst(serlaplace(p),y,1) | p<-Vec(prod(k=1, n, subst(q + O(x*x^(n\k)), x, x^k)))]} \\ _Andrew Howroyd_, Feb 12 2022
%Y A351013 The version for run-lengths instead of runs is A329739, normal A329740.
%Y A351013 These compositions are ranked by A351290, complement A351291.
%Y A351013 A000005 counts constant compositions, ranked by A272919.
%Y A351013 A005811 counts runs in binary expansion.
%Y A351013 A011782 counts integer compositions.
%Y A351013 A059966 counts binary Lyndon compositions, necklaces A008965, aperiodic A000740.
%Y A351013 A116608 counts compositions by number of distinct parts.
%Y A351013 A238130 and A238279 count compositions by number of runs.
%Y A351013 A242882 counts compositions with distinct multiplicities.
%Y A351013 A297770 counts distinct runs in binary expansion.
%Y A351013 A325545 counts compositions with distinct differences.
%Y A351013 A329744 counts compositions by runs-resistance.
%Y A351013 A351014 counts distinct runs in standard compositions.
%Y A351013 Counting words with all distinct runs:
%Y A351013 - A351016 = binary words, for run-lengths A351017.
%Y A351013 - A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
%Y A351013 - A351200 = patterns, for run-lengths A351292.
%Y A351013 - A351202 = permutations of prime factors.
%Y A351013 Cf. A003242, A025047, A044813, A098504, A098859, A106356, A329738, A328592, A334028, A351015, A351201, A351204.
%K A351013 nonn
%O A351013 0,3
%A A351013 _Gus Wiseman_, Feb 09 2022
%E A351013 Terms a(26) and beyond from _Andrew Howroyd_, Feb 12 2022