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Various formulas counting the number of permutations:
A000522(n) = A038155(n+2)/A000217(n+1).
A052801(n) = A007840(n) + A215916(n)
A067318(n) + A001705(n-1) = A062119(n)
Several of them have been found empirically from this useful configuration:
http://imageshack.com/a/img924/259/i36XVh.jpg
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