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A335872
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Number T(n,k) of permutations of [n] having k points that are fixed or reflected; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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3
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1, 0, 1, 0, 0, 2, 0, 4, 0, 2, 4, 0, 16, 0, 4, 16, 36, 32, 32, 0, 4, 80, 192, 216, 128, 96, 0, 8, 672, 1472, 1440, 984, 320, 144, 0, 8, 4752, 10752, 11776, 7680, 3936, 1024, 384, 0, 16, 48768, 103568, 104448, 65920, 28544, 9312, 1792, 512, 0, 16
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OFFSET
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0,6
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COMMENTS
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A permutation p of [n] has fixed point j if p(j) = j, it has reflected point j if p(n+1-j) = j. A point can be fixed and reflected at the same time.
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LINKS
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FORMULA
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Sum_{k=1..n} k * T(n,k) = A335873(n).
T(n,n-2) = floor((n-1)^2/2) * 2^floor(n/2).
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EXAMPLE
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1;
0, 1;
0, 0, 2;
0, 4, 0, 2;
4, 0, 16, 0, 4;
16, 36, 32, 32, 0, 4;
80, 192, 216, 128, 96, 0, 8;
672, 1472, 1440, 984, 320, 144, 0, 8;
4752, 10752, 11776, 7680, 3936, 1024, 384, 0, 16;
48768, 103568, 104448, 65920, 28544, 9312, 1792, 512, 0, 16;
...
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MAPLE
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b:= proc(s, i, t) option remember; (n-> `if`(n=0, x^t, add(
b(s minus {j}, i+1, t+`if`(j in {i, n}, 1, 0)), j=s)))(nops(s))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b({$1..n}, 1, 0)):
seq(T(n), n=0..10);
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MATHEMATICA
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b[s_, i_, t_] := b[s, i, t] = With[{n = Length[s]}, If[n == 0, x^t, Sum[b[s ~Complement~ {j}, i+1, t + If[j == i || j == n, 1, 0]], {j, s}]]];
T[n_] := CoefficientList[b[Range[n], 1, 0], x];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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