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Number of permutations that avoid the consecutive pattern 1423.
7

%I #41 Oct 27 2023 21:38:02

%S 1,1,2,6,23,110,631,4218,32221,276896,2643883,27768955,318174363,

%T 3949415431,52794067318,756137263377,11551672922816,187507250145806,

%U 3222662529113641,58464560588277289,1116469710152742025,22386721651323946628,470259350616967829363

%N Number of permutations that avoid the consecutive pattern 1423.

%H Alois P. Heinz, <a href="/A201692/b201692.txt">Table of n, a(n) for n = 0..250</a> (terms n = 0..60 from Ray Chandler)

%H A. Baxter, B. Nakamura, and D. Zeilberger. <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/auto.html">Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes</a>

%H V. Dotsenko and A. Khoroshkin, <a href="http://arxiv.org/abs/1109.2690">Shuffle algebras, homology, and consecutive pattern avoidance</a>, arXiv preprint arXiv:1109.2690, 2011

%F 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.

%F a(n) ~ c * d^n * n!, where d = 0.95482605094987833345080179991528996596888600981..., c = 1.1567436851576902067739566662625378535625602... . - _Vaclav Kotesovec_, Sep 11 2014

%p c := proc(n,l)

%p if n = 1 then

%p if l = 0 then

%p 1;

%p else

%p 0;

%p end if;

%p elif n= 2 or n = 3 then

%p 0;

%p else

%p a := 0 ;

%p for k from 1 to (n-2)/2 do

%p a := a+procname(n-2*k-1,l-k)*binomial(n-k-2,k) ;

%p end do:

%p a ;

%p end if;

%p end proc:

%p A201693 := proc(nmax)

%p g := 1-t ;

%p for n from 2 to nmax do

%p for l from 0 to n/2 do

%p g := g-c(n,l)*t^n*(-1)^l/n! ;

%p end do:

%p end do:

%p taylor(1/g,t=0,nmax) ;

%p end proc:

%p nmax := 25 ;

%p egf := A201693(nmax) ;

%p for n from 0 to nmax-1 do

%p printf("%d,",coeftayl(egf,t=0,n)*n!) ;

%p end do: # _R. J. Mathar_, Dec 04 2011

%p # second Maple program:

%p b:= proc(u, o, t) option remember; `if`(u+o=0, 1,

%p add(b(u-j, o+j-1, `if`(0<t and j<t, -j, 0)), j=1..u)+

%p add(b(u+j-1, o-j, j), j=`if`(t<0, -t, 1)..o))

%p end:

%p a:= n-> b(n, 0$2):

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Nov 07 2013

%t 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_ *)

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

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, Dec 03 2011

%E Definition corrected by _N. J. A. Sloane_, Mar 15 2015