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A064314 Total length of longest increasing runs in all permutations of n elements.

%I A064314 #21 Apr 22 2022 04:57:04
%S A064314 1,3,12,55,299,1900,13942,115932,1078361,11092265,125040100,
%T A064314 1532995992,20310212672,289186696338,4404156016584,71441907922793,
%U A064314 1229835421590959,22393298253477006,430019590699868644,8685717780508953928,184088653170341473400,4085097253151506682170
%N A064314 Total length of longest increasing runs in all permutations of n elements.
%H A064314 Alois P. Heinz, <a href="/A064314/b064314.txt">Table of n, a(n) for n = 1..200</a> (first 100 terms from Max Alekseyev)
%F A064314 a(n) = Sum_{k=1..n} k*A008304(n,k). - _Max Alekseyev_, May 22 2012
%p A064314 b:= proc(u, o, t, k) option remember; `if`(t=k, (u+o)!,
%p A064314       `if`(max(t, u)+o<k, 0, add(b(u+j-1, o-j, t+1, k), j=1..o)+
%p A064314       add(b(u-j, o+j-1, 1, k), j=1..u)))
%p A064314     end:
%p A064314 a:= n-> add(b(0, n, 0, k), k=1..n) -n*b(0, n, 0, n+1):
%p A064314 seq(a(n), n=1..25);  # _Alois P. Heinz_, Oct 16 2013
%t A064314 nn=30;f[list_]:=Total[Table[list[[i]]*i,{i,1,Length[list]}]];a[r_]:=Apply[Plus,Table[Normal[Series[y x^(r+1)/(1-Sum[y x^i,{i,1,r}]),{x,0,nn}]][[n]]/(n+r)!,{n,1,nn-r}]]/.y->-1;Map[f,Map[Select[#,#>0&]&,Transpose[Prepend[Table[Drop[Range[0,nn]! CoefficientList[Series[1/(1-x-a[n+1])-1/(1-x-a[n]),{x,0,nn}],x],1],{n,1,28}],Table[1,{nn}]]]]] (* _Geoffrey Critzer_, Feb 25 2014 *)
%Y A064314 This sequence treats runs of adjacent elements, A003316 treats subsequences of not necessarily adjacent elements.
%K A064314 nonn
%O A064314 1,2
%A A064314 _David W. Wilson_, Sep 07 2001