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A173939
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The number of permutations avoiding simultaneously consecutive patterns 132 and 213.
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1
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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
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OFFSET
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0,3
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LINKS
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FORMULA
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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
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EXAMPLE
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Example: For n = 3 a(3) = 4 since 123, 231, 312, and 321 are the 3-permutations avoiding 132 and 213.
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MAPLE
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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):
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MATHEMATICA
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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];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Signy Olafsdottir (signy06(AT)ru.is), Mar 03 2010
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EXTENSIONS
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STATUS
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approved
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