A158432 Number of permutations of 1..n containing the relative rank sequence { 45312 } at any spacing.

1, 26, 458, 6996, 101072, 1438112, 20598112, 300892896, 4521034917, 70286670034, 1135485759114, 19121776482564, 336412530327804, 6191800556586104, 119301546930406184, 2406376964044265344, 50786085223779295344, 1120447461653440780128, 25810064637612342838624

Offset: 5

Author

R. H. Hardin, Mar 18 2009

Keywords

• nonn

Comments

Same series for 54321 12345 45321 21345 12354 54312 34521 32145 12543 54123 23451 43215 15432 51234 21354 34512 32154 21543 45123.

Links

• Alois P. Heinz, Table of n, a(n) for n = 5..200
• Eric Weisstein's World of Mathematics, Permutation Pattern

Crossrefs

Cf. A000142, A047889, A056986, A158005, A214015, A214152.

Programs

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:= (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))):
a:= n-> n! -g(n, 4, []):
seq(a(n), n=5..25);  # Alois P. Heinz, Jul 05 2012
# second Maple program
a:= proc(n) option remember; `if`(n<5, 0, `if`(n=5, 1,
     ((132-142*n-301*n^2-35*n^3+25*n^4+n^5)*a(n-1)
     -2*(10*n^3+33*n^2-181*n-2)*(n-1)^2*a(n-2)
     +64*(n-2)^2*(n-1)^3*a(n-3))/ ((n+4)*(n-5)*(n+3)^2)))
    end:
seq(a(n), n=5..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, 4, {}];
Table[a[n], {n, 5, 25}] (* Jean-François Alcover, Jun 19 2018, after Alois P. Heinz's first program *)

Formula

a(n) = A214152(n,5) = A000142(n)-A047889(n) = A000142(n)-A214015(n,4).

Extensions

Extended beyond a(16) by Alois P. Heinz, Jul 05 2012