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 A384891 #13 Jun 22 2025 15:13:33
%S A384891 1,1,1,3,3,5,23,25,43,63,345,365,665,949,1513,8175,9003,15929,23399,
%T A384891 36949,51043,293715,314697,570353,826817,1318201,1810393,2766099,
%U A384891 14180139,15600413,27707879,40501321,63981955,88599903,134362569,181491125,923029217
%N A384891 Number of permutations of {1..n} with all distinct lengths of maximal runs (increasing by 1).
%H A384891 Christian Sievers, <a href="/A384891/b384891.txt">Table of n, a(n) for n = 0..5000</a>
%F A384891 a(n) = Sum_{k=1..n} ( T(n,k) * A000255(k-1) ) for n>=1, where T(n,k) is the number of compositions of n into k distinct parts (cf. A072574). - _Christian Sievers_, Jun 22 2025
%e A384891 The permutation (1,2,6,7,8,9,3,4,5) has maximal runs ((1,2),(6,7,8,9),(3,4,5)), with lengths (2,4,3), so is counted under a(9).
%e A384891 The a(0) = 1 through a(7) = 25 permutations:
%e A384891 () (1) (12) (123) (1234) (12345) (123456) (1234567)
%e A384891 (231) (2341) (23451) (123564) (1234675)
%e A384891 (312) (4123) (34512) (123645) (1234756)
%e A384891 (45123) (124563) (1245673)
%e A384891 (51234) (126345) (1273456)
%e A384891 (145623) (1456723)
%e A384891 (156234) (1672345)
%e A384891 (231456) (2314567)
%e A384891 (234156) (2345167)
%e A384891 (234561) (2345671)
%e A384891 (312456) (3124567)
%e A384891 (345126) (3456127)
%e A384891 (345612) (3456712)
%e A384891 (412356) (4567123)
%e A384891 (451236) (4567231)
%e A384891 (456231) (4567312)
%e A384891 (456312) (5123467)
%e A384891 (561234) (5612347)
%e A384891 (562341) (5671234)
%e A384891 (564123) (6712345)
%e A384891 (612345) (6723451)
%e A384891 (634512) (6751234)
%e A384891 (645123) (7123456)
%e A384891 (7345612)
%e A384891 (7561234)
%t A384891 Table[Length[Select[Permutations[Range[n]],UnsameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}]
%o A384891 (PARI) lista(n)=my(b(n)=sum(i=0,n-1,(-1)^i*(n-i)!*binomial(n-1,i)), d=floor(sqrt(2*n)), p=prod(i=1,n,1+x*y^i,1+O(y*y^n)*((1-x^(n+1))/(1-x))+O(x*x^d))); Vec(1+sum(i=1,d,i!*b(i)*polcoef(p,i))) \\ _Christian Sievers_, Jun 22 2025
%Y A384891 Counting by number of maximal anti-runs gives A010027, for runs A123513.
%Y A384891 For subsets instead of permutations we have A384175, complement A384176.
%Y A384891 For partitions we have A384884 (anti-runs A384885), strict A384178 (anti-runs A384880).
%Y A384891 For equal instead of distinct lengths we have A384892.
%Y A384891 For anti-runs instead of runs we have A384907.
%Y A384891 A000041 counts integer partitions, strict A000009.
%Y A384891 A034839 counts subsets by number of maximal runs, for strict partitions A116674.
%Y A384891 A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
%Y A384891 A356606 counts strict partitions without a neighborless part, complement A356607.
%Y A384891 A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
%Y A384891 Cf. A000255, A044813, A072574, A242882, A287170, A325324, A325325, A328592, A329739, A351202, A384177, A384886.
%K A384891 nonn
%O A384891 0,4
%A A384891 _Gus Wiseman_, Jun 19 2025
%E A384891 a(11) and beyond from _Christian Sievers_, Jun 22 2025