cp's OEIS Frontend

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

%I A114209 #9 Mar 13 2025 11:10:00
%S A114209 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,
%T A114209 40,35,45,40,51,45,57,51,63,57,70,63,77,70,84,77,92,84,100,92,108,100,
%U A114209 117,108,126,117,135,126,145,135,155,145,165,155,176,165,187,176,198,187
%N A114209 Number of permutations of [n] having exactly two fixed points and avoiding the patterns 123 and 231.
%H A114209 T. Mansour and A. Robertson, <a href="http://dx.doi.org/10.1007/s000260200013">Refined restricted permutations avoiding subsets of patterns of length three</a>, Annals of Combinatorics, 6, 2002, 407-418.
%H A114209 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,1,-1,-2,0,1).
%F A114209 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.
%F A114209 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]
%e A114209 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.
%p A114209 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);
%Y A114209 Cf. A008731, A114208, A114210.
%K A114209 nonn
%O A114209 1,4
%A A114209 _Emeric Deutsch_, Nov 17 2005