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A114208 Number of permutations of [n] having exactly one fixed point and avoiding the patterns 123 and 231. 3
1, 0, 3, 2, 6, 6, 12, 10, 21, 16, 31, 24, 44, 32, 60, 42, 77, 54, 97, 66, 120, 80, 144, 96, 171, 112, 201, 130, 232, 150, 266, 170, 303, 192, 341, 216, 382, 240, 426, 266, 471, 294, 519, 322, 570, 352, 622, 384, 677, 416, 735, 450, 794, 486, 856, 522, 921, 560 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

REFERENCES

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

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (-1,2,3,0,-3,-2,1,1).

FORMULA

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.

a(1)=1, a(2)=0, a(3)=3, a(4)=2, a(5)=6, a(6)=6, a(7)=12, a(8)=10, 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) [From Harvey P. Dale, Mar 05 2012]

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]

EXAMPLE

a(2)=0 because none of the permutations 12 and 21 has exactly one fixed point;

a(3)=3 because we have 132, 213 and 321; a(4)=2 because we have 4132 and 4213.

MAPLE

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);

MATHEMATICA

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 *)

CROSSREFS

Cf. A114209, A114210.

Sequence in context: A157793 A096375 A062200 * A014686 A053090 A264400

Adjacent sequences:  A114205 A114206 A114207 * A114209 A114210 A114211

KEYWORD

nonn,easy

AUTHOR

Emeric Deutsch, Nov 17 2005

STATUS

approved

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Last modified December 5 17:28 EST 2021. Contains 349557 sequences. (Running on oeis4.)