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A114208 Number of permutations of [n] having exactly one fixed point and avoiding the patterns 123 and 231.

%I A114208 #16 Mar 27 2024 08:58:46
%S A114208 1,0,3,2,6,6,12,10,21,16,31,24,44,32,60,42,77,54,97,66,120,80,144,96,
%T A114208 171,112,201,130,232,150,266,170,303,192,341,216,382,240,426,266,471,
%U A114208 294,519,322,570,352,622,384,677,416,735,450,794,486,856,522,921,560
%N A114208 Number of permutations of [n] having exactly one fixed point and avoiding the patterns 123 and 231.
%H A114208 Harvey P. Dale, <a href="/A114208/b114208.txt">Table of n, a(n) for n = 1..1000</a>
%H A114208 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 A114208 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (-1,2,3,0,-3,-2,1,1).
%F A114208 n^2/6 if n mod 6 = 0; (7*n^2-12*n+29)/24 if n mod 6 = 1 or 5; (n^2-4)/6 if n mod 6 = 2 or 4; (7*n^2-12*n+45)/24 if n mod 6 = 3.
%F A114208 a(n) = a(n-1)+ 2*a(n-2)+3*a(n-3)-3*a(n-5)-2*a(n-6)+a(n-7)+a(n-8). [_Harvey P. Dale_, Mar 05 2012]
%F A114208 G.f.: -x*(2*x^6+2*x^5+2*x^4+2*x^3+x^2+x+1) / ((x-1)^3*(x+1)^3*(x^2+x+1)). [_Colin Barker_, Aug 11 2013]
%e A114208 a(2)=0 because none of the permutations 12 and 21 has exactly one fixed point.
%e A114208 a(3)=3 because we have 132, 213 and 321.
%e A114208 a(4)=2 because we have 4132 and 4213.
%p A114208 a:=proc(n) if n mod 6 = 0 then n^2/6 elif n mod 6 = 1 or n mod 6 = 5 then (7*n^2-12*n+29)/24 elif n mod 6 = 2 or n mod 6 = 4 then (n^2-4)/6 else (7*n^2-12*n+45)/24 fi end: seq(a(n),n=1..70);
%t A114208 npn[n_]:=Module[{c=Mod[n,6]},Which[c==0,n^2/6,c==1,(7n^2-12n+29)/24,c==2,(n^2-4)/6,c==3,(7n^2-12n+45)/24,c==4,(n^2-4)/6,c==5,(7n^2-12n+29)/24]]; Array[npn,60] (* or *) LinearRecurrence[{-1,2,3,0,-3,-2,1,1},{1,0,3,2,6,6,12,10},60] (* _Harvey P. Dale_, Mar 05 2012 *)
%Y A114208 Cf. A114209, A114210.
%K A114208 nonn,easy
%O A114208 1,3
%A A114208 _Emeric Deutsch_, Nov 17 2005