A114209 Number of permutations of [n] having exactly two fixed points and avoiding the patterns 123 and 231.

0, 1, 0, 2, 1, 3, 2, 5, 3, 7, 5, 9, 7, 12, 9, 15, 12, 18, 15, 22, 18, 26, 22, 30, 26, 35, 30, 40, 35, 45, 40, 51, 45, 57, 51, 63, 57, 70, 63, 77, 70, 84, 77, 92, 84, 100, 92, 108, 100, 117, 108, 126, 117, 135, 126, 145, 135, 155, 145, 165, 155, 176, 165, 187, 176, 198, 187

Offset

1,4

Links

Table of n, a(n) for n=1..67.

T. Mansour and A. Robertson, Refined restricted permutations avoiding subsets of patterns of length three, Annals of Combinatorics, 6, 2002, 407-418.

Formula

a(n) = n(n+6)/24 if n mod 6 = 0; (n^2-1)/24 if n mod 6 = 1 or 5; (n+2)(n+4)/24 if n mod 6 = 2 or 4; (n^2-9)/24 if n mod 6 = 3.

a(n) = A008731(n-2). O.g.f.: x^2/((1-x)^3(1+x)^2(1+x+x^2)). [R. J. Mathar, Aug 11 2008]

Example

a(2)=1 because we have 12; a(3)=0 because no permutation of [3] can have exactly two fixed points; a(4)=2 because we have 1432 and 3214.

Maple

a:=proc(n) if n mod 6 = 0 then n*(n+6)/24 elif n mod 6 = 1 or n mod 6 = 5 then (n^2-1)/24 elif n mod 6 = 2 or n mod 6 = 4 then (n+2)*(n+4)/24 else (n^2-9)/24 fi end: seq(a(n), n=1..70);

Crossrefs

Cf. A008731, A114208, A114210.

Sequence in context: A161051 A161255 A008731 * A132091 A262090 A239881

Adjacent sequences: A114206 A114207 A114208 * A114210 A114211 A114212

Keyword

nonn

Author

Emeric Deutsch, Nov 17 2005

Status

approved