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

0,3

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

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

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<t and j<t, -j, 0)), j=1..u)+
      add(b(u+j-1, o-j, j), j=`if`(t<0, -t, 1)..o))
    end:
a:= n-> 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[0<t && j<t, -j, 0]], {j, 1, u}] + Sum[b[u+j-1, o-j, j], {j, If[t<0, -t, 1], o}]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 18 2014, after Alois P. Heinz *)

Crossrefs

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

Sequence in context: A205802 A117226 A117156 * A113229 A113228 A201693

Adjacent sequences: A201689 A201690 A201691 * A201693 A201694 A201695

Keyword

nonn,easy

Author

N. J. A. Sloane, Dec 03 2011

Extensions

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

Status

approved