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A173939 The number of permutations avoiding simultaneously consecutive patterns 132 and 213. 1
1, 1, 2, 4, 11, 37, 149, 705, 3814, 23199, 156940, 1167862, 9478482, 83347221, 789272024, 8007691756, 86661018861, 996483990245, 12132147428205, 155914784843610, 2109180000370399, 29959103479924280, 445807660125129870, 6935399147203692664, 112584529786912528601 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
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.
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
Sequence in context: A281481 A138301 A367495 * A328433 A118182 A179327
KEYWORD
nonn
AUTHOR
Signy Olafsdottir (signy06(AT)ru.is), Mar 03 2010
EXTENSIONS
More terms from Alois P. Heinz, Nov 03 2013
STATUS
approved

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Last modified March 29 10:59 EDT 2024. Contains 371277 sequences. (Running on oeis4.)