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

1, 3, 12, 55, 299, 1900, 13942, 115932, 1078361, 11092265, 125040100, 1532995992, 20310212672, 289186696338, 4404156016584, 71441907922793, 1229835421590959, 22393298253477006, 430019590699868644, 8685717780508953928, 184088653170341473400, 4085097253151506682170

Offset: 1

Author

David W. Wilson, Sep 07 2001

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..200 (first 100 terms from Max Alekseyev)

Crossrefs

This sequence treats runs of adjacent elements, A003316 treats subsequences of not necessarily adjacent elements.

Programs

Maple

b:= proc(u, o, t, k) option remember; `if`(t=k, (u+o)!,
      `if`(max(t, u)+o add(b(0, n, 0, k), k=1..n) -n*b(0, n, 0, n+1):
seq(a(n), n=1..25);  # Alois P. Heinz, Oct 16 2013

Mathematica

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 *)

Formula

a(n) = Sum_{k=1..n} k*A008304(n,k). - Max Alekseyev, May 22 2012