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 A327833 #16 Oct 23 2019 16:40:49
%S A327833 1,1,4,18,100,665,5124,44772,437016,4710915,55568480,711802894,
%T A327833 9838192572,145921265581,2311617527660,38950657146120,695562375445104,
%U A327833 13121344429311687,260728755911619336,5443039353326333330,119101575356825879860,2725785134463572716689
%N A327833 Number of non-overlapping pairs of adjacent runs in permutations of [n].
%C A327833 A run is a maximal consecutive subsequence of increasing values; two adjacent runs are non-overlapping if the least value in the first run exceeds the greatest value in the second.
%C A327833 Permutations all of whose adjacent runs overlap are in the image of the pop-stack sorting operation (see A307030 and references).
%F A327833 a(n) = (n-1)*n! - (n/3-1/2)*floor(e*n!) + (n/6-1/2), for all n > 1.
%F A327833 Asymptotically, the expected number of non-overlapping adjacent pairs of runs an n-permutation is (1-e/3)*n + (e/2-1).
%e A327833 a(3) = 4: one non-overlapping pair of adjacent runs in both 231 and 312, and two non-overlapping pairs in 321; the pairs of adjacent runs in 132 and 213 overlap.
%t A327833 Table[If[n==1,1,(n-1)n!-(n/3-1/2)Floor[E n!]+(n/6-1/2)],{n,20}]
%K A327833 nonn
%O A327833 1,3
%A A327833 _David Bevan_, Sep 27 2019