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A161130
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Sum of the differences between the largest and the smallest fixed points over all non-derangement permutations of {1,2,...,n}.
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1
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0, 0, 1, 2, 13, 74, 523, 4178, 37609, 376082, 4136911, 49642922, 645357997, 9035011946, 135525179203, 2168402867234, 36862848742993, 663531277373858, 12607094270103319, 252141885402066362, 5294979593443393621
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| a(n)=A000166(n+1)-A155521(n).
a(n)=Sum(k*A161129(n,k),k=0..n-1).
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REFERENCES
| E. Deutsch and S. Elizalde, The largest and the smallest fixed points of permutations, arXiv:0904.2792v1, 2009.
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FORMULA
| E.g.f.: G=[exp(-x)*(1+x+x^2) - 1]/(1-x)^2.
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EXAMPLE
| a(3)=2 because the non-derangements of {1,2,3} are 1'23', 1'32, 213', and 32'1 with differences between the largest and smallest fixed points (marked) equal to 2, 0, 0, and 0, respectively.
a(4)=13 because the non-derangements of {1,2,3,4} are 1'234', 1'2'43, 1'423, 1'324', 1'342, 1'43'2, 413'2, 3124', 213'4', 42'13, 2314', 243'1, 42'3'1, 32'14', and 32'41 with differences between the largest and smallest fixed points (marked) equal to 3, 1, 0, 3, 0, 2, 0, 0, 1, 0, 0, 0, 1, 2, and 0, respectively.
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MAPLE
| G := (exp(-x)*(1+x+x^2)-1)/(1-x)^2: Gser := series(G, x = 0, 25): seq(factorial(n)*coeff(Gser, x, n), n = 0 .. 22);
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CROSSREFS
| A000166, A155521, A161129
Sequence in context: A163190 A004027 A154357 * A192700 A007509 A077413
Adjacent sequences: A161127 A161128 A161129 * A161131 A161132 A161133
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 18 2009
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