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A130153
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For each permutation p of {1,2,...,n} define maxabsjump(p) = max(|p(i) - i|, 1<=i<=n); a(n) is the sum of maxabsjumps of all p.
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4
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0, 1, 8, 52, 362, 2779, 23749, 224570, 2334827, 26504418, 326476313, 4338953453, 61908434299, 944065986251, 15325676747363, 263910466293264, 4805394642545408, 92254524334410750, 1862526899218400010, 39449927915059031609, 874745258339527435041, 20265180489604116160763
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OFFSET
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1,3
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COMMENTS
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a(n) = Sum_{k=0..n-1} k*A130152(n,k). For the case when in the definition |p(i)-i| is replaced by p(i)-i, see A018927.
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LINKS
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EXAMPLE
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a(3) = 8 because the permutations 123,132,213,231,312 and 321 have maxabsjumps 0,1,1,2,2 and 2, respectively.
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MAPLE
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with(combinat): for n from 1 to 7 do P:=permute(n): for i from 0 to n-1 do ct[i]:=0 od: for j from 1 to n! do if max(seq(abs(P[j][i]-i), i=1..n))=0 then ct[0]:=ct[0]+1 elif max(seq(abs(P[j][i]-i), i=1..n))=1 then ct[1]:=ct[1]+1 elif max(seq(abs(P[j][i]-i), i=1..n))=2 then ct[2]:=ct[2]+1 elif max(seq(abs(P[j][i]-i), i=1..n))=3 then ct[3]:=ct[3]+1 elif max(seq(abs(P[j][i]-i), i=1..n))=4 then ct[4]:=ct[4]+1 elif max(seq(abs(P[j][i]-i), i=1..n))=5 then ct[5]:=ct[5]+1 elif max(seq(abs(P[j][i]-i), i=1..n))=6 then ct[6]:=ct[6]+1 else fi od: a[n]:=sum(k*ct[k], k=0..n-1): od: seq(a[n], n=1..7); # a cumbersome program to obtain the first 7 terms of the sequence
n := 8: st := proc (p) max(seq(abs(p[j]-j), j = 1 .. nops(p))) end proc: with(combinat): P := permute(n): f := sort(add(t^st(P[i]), i = 1 .. factorial(n))): subs(t = 1, diff(f, t)); # program yields a(n) for the specified n - Emeric Deutsch, Aug 13 2009
# second Maple program:
b:= proc(s) option remember; (n-> `if`(n=0, 1, add((p-> add(
coeff(p, x, i)*x^max(i, abs(n-j)), i=0..degree(p)))(
b(s minus {j})), j=s)))(nops(s))
end:
a:= n-> (p-> add(coeff(p, x, i)*i, i=1..n-1))(b({$1..n})):
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MATHEMATICA
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b[s_] := b[s] = Function[n, If[n == 0, 1, Sum[Function[p, Sum[Coefficient[p, x, i] x^Max[i, Abs[n-j]], {i, 0, Exponent[p, x]}]][b[s ~Complement~ {j}]], {j, s}]]][Length[s]];
a[n_] := Function[p, Sum[Coefficient[p, x, i] i, {i, 1, n-1}]][b[Range[n]]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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