OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..300
R. E. L. Aldred, M. D. Atkinson, and D. J. McCaughan, Avoiding consecutive patterns in permutations Adv. in Appl. Math., 45(3), 449-461, 2010.
A. Baxter, B. Nakamura, and D. Zeilberger, Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes
S. Kitaev, Multi-avoidance of Generalised Patterns
S. Kitaev and T. Mansour, On multi-avoidance of generalized patterns, arXiv:math/0209340 [math.CO], 2002; Ars Combinatoria 76 (2005), 321-350.
FORMULA
We have
a(n) = Sum_{i=1..n, j=1..n} b(n;i,j) for n > 1 with
b(n;i,i) = 0 for all n, i >= 1;
b(n;i,j) = Sum_{k=1..i-1} b(n-1;j,k) if i > j;
b(n;i,j) = Sum_{k=1..i-1} b(n-1;j-1,k) + Sum_{k=j..n-1} b(n-1;j-1,k) if i < j;
except for the cases
b(2;1,2) = b(2;2,1) = 1.
a(n) ~ c * d^n * n!, where d = 0.6763882280940348940704054..., c = 2.158082675431352391418684... . - Vaclav Kotesovec, Aug 22 2014
EXAMPLE
Example: For n = 3 a(3) = 4 since 123, 231, 312, and 321 are the 3-permutations avoiding 132 and 213.
MAPLE
g:= proc(u, o, t) option remember; `if`(u+o=0, 1,
add(g(u-j, o+j-1, -j), j=`if`(t>0, t, 1)..u)+
add(g(u+j-1, o-j, +j), j=1..`if`(t<0, -t-1, o)))
end:
a:= n-> g(n, 0$2):
seq(a(n), n=0..26); # Alois P. Heinz, Nov 04 2013
MATHEMATICA
b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, Sum[b[u - j, o + j - 1, -j], {j, If[t > 0, t, 1], u}] + Sum[b[u + j - 1, o - j, +j], {j, 1, If[t < 0, -t - 1, o]}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 21 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Signy Olafsdottir (signy06(AT)ru.is), Mar 03 2010
EXTENSIONS
More terms from Alois P. Heinz, Nov 03 2013
STATUS
approved