A159139 Number of permutations of 1..n containing the relative rank sequence { 213465 } at any spacing.

1, 37, 891, 18043, 337210, 6081686, 108469917, 1941309261, 35187952132, 649951312000, 12286366975723, 238445927000811, 4762398793018878, 98074791689121162, 2085684931155975120, 45859509146309390064, 1043533983233372354613, 24590543663448304800169

Offset

6,2

Comments

Same series for 654321 123456 564321 213456 123465 654312 456321 321456 123654 654123 345621 432156 126543 651234 564312 456312 321465 213654 564123 345612 432165 216543 561234 234561 543216 165432 612345 456123 321654.

Links

Alois P. Heinz, Table of n, a(n) for n = 6..200

Eric Weisstein's World of Mathematics, Permutation Pattern

Formula

a(n) = A214152(n,6) = A000142(n)-A047890(n) = A000142(n)-A214015(n,5). - Alois P. Heinz, Jul 05 2012

Maple

h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
      +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l)
      `if`(n=0 or i=1, h([l[], 1$n])^2, `if`(i<1, 0,
       add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
    end:
a:= n-> n! -g(n, 5, []):
seq(a(n), n=6..30);  # Alois P. Heinz, Jul 05 2012
# second Maple program
a:= proc(n) option remember; `if`(n<6, 0, `if`(n=6, 1,
     ((2475-4819*n^2-2985*n+175*n^4-1021*n^3+n^6+49*n^5)*a(n-1)
     -(35*n^4+441*n^3-845*n^2-4147*n-489)*(n-1)^2*a(n-2)
     +(-1668+329*n+259*n^2)*(n-1)^2*(n-2)^2*a(n-3)
     -225*(n-1)^2*(n-2)^2*(n-3)^2*a(n-4))/ ((n-6)*(n+6)^2*(n+4)^2)))
    end:
seq(a(n), n=6..30);  # Alois P. Heinz, Sep 26 2012

Mathematica

h[l_] := With[{n = Length[l]}, Sum[i, {i, l}]!/Product[Product[1+l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := If[n == 0 || i === 1, h[Join[l, Array[1 &, n]]]^2, If[i < 1, 0, Sum[g[n - i*j, i - 1, Join[l, Array[i &, j]]], {j, 0, n/i}]]];
a[n_] := n! - g[n, 5, {}];
Table[a[n], {n, 6, 30}] (* Jean-François Alcover, Jun 19 2018, from first Maple program *)

Crossrefs

Cf. A000142, A047890, A056986, A158005, A158423, A214015, A214152.

Sequence in context: A159127 A159141 A159089 * A159091 A159140 A159132

Adjacent sequences: A159136 A159137 A159138 * A159140 A159141 A159142

Keyword

nonn

Author

R. H. Hardin, Apr 05 2009

Extensions

More terms from Alois P. Heinz, Jul 05 2012

Status

approved