A201692 Number of permutations that avoid the consecutive pattern 1423.

1, 1, 2, 6, 23, 110, 631, 4218, 32221, 276896, 2643883, 27768955, 318174363, 3949415431, 52794067318, 756137263377, 11551672922816, 187507250145806, 3222662529113641, 58464560588277289, 1116469710152742025, 22386721651323946628, 470259350616967829363

Offset: 0

Author

N. J. A. Sloane, Dec 03 2011

Keywords

• nonn
• easy

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..250 (terms n = 0..60 from Ray Chandler)
• A. Baxter, B. Nakamura, and D. Zeilberger. Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes
• V. Dotsenko and A. Khoroshkin, Shuffle algebras, homology, and consecutive pattern avoidance, arXiv preprint arXiv:1109.2690, 2011

Crossrefs

Cf. A113228, A113229, A117156, A117158, A117226, A201693.

Programs

Maple

c := proc(n,l)
    if n = 1 then
        if l = 0 then
            1;
        else
            0;
        end if;
    elif n= 2 or n = 3 then
        0;
    else
        a := 0 ;
        for k from 1 to (n-2)/2 do
            a := a+procname(n-2*k-1,l-k)*binomial(n-k-2,k) ;
        end do:
        a ;
    end if;
end proc:
A201693 := proc(nmax)
    g := 1-t ;
    for n from 2 to nmax do
        for l from 0 to n/2 do
            g := g-c(n,l)*t^n*(-1)^l/n! ;
        end do:
    end do:
    taylor(1/g,t=0,nmax) ;
end proc:
nmax := 25 ;
egf := A201693(nmax) ;
for n from 0 to nmax-1 do
    printf("%d,",coeftayl(egf,t=0,n)*n!) ;
end do: # R. J. Mathar, Dec 04 2011
# second Maple program:
b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      add(b(u-j, o+j-1, `if`(0 b(n, 0$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 07 2013

Mathematica

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Sum[b[u-j, o+j-1, If[0Jean-François Alcover, Mar 18 2014, after Alois P. Heinz *)

Formula

The reference gives an e.g.f. There is an associated triangle of numbers c_{n,l} that should be added to the OEIS if it is not already present.

a(n) ~ c * d^n * n!, where d = 0.95482605094987833345080179991528996596888600981..., c = 1.1567436851576902067739566662625378535625602... . - Vaclav Kotesovec, Sep 11 2014

Extensions

Definition corrected by N. J. A. Sloane, Mar 15 2015