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Search: id:A090672
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| 0, 2, 16, 120, 960, 8400, 80640, 846720, 9676800, 119750400, 1596672000, 22832409600, 348713164800, 5666588928000, 97639686144000, 1778437140480000, 34145993097216000, 689322235650048000, 14597412049059840000, 323575967087493120000, 7493338185184051200000
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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a(n) = Sum_{pi in Symm(n)} Sum_{i=1..n} |pi(i)-i|, i.e. the total displacement of all letters in all permutations on n letters.
a(n)=number of entries between the entries 1 and 2 in all permutations of {1,2,...,n+1}. Example: a(2)=2 because we have 123, 1(3)2, 213, 2(3)1, 312, 321; the entries between 1 and 2 are surrounded by parentheses. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 06 2008
a(n)=Sum(k*A138770(n+1,k),k=0..n-1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 06 2008
Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 26 2009: (Start)
a(n) is also the number of peaks in all permutations of {1,2,...,n+1}. Example: a(3)=16 because the permutations 1234, 4123, 3124, 4312, 2134, 4213, 3214, and 4321 have no peaks and each of the remaining 16 permutations of {1,2,3,4} has exactly one peak.
(End)
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REFERENCES
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D. Daly and P. Vojtechovsky, Displacement of permutations, preprint, 2003.
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FORMULA
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a(n)=A052571(n+2)/3 =2*A005990(n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 11 2007
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MAPLE
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a:=n->sum(sum(sum(n!/3, j=1..n), k=-1..n), m=0..n): seq(a(n), n=0..20); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 11 2007
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CROSSREFS
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Twice A005990.
Cf. A138770.
Sequence in context: A026129 A026158 A025185 this_sequence A027309 A069868 A022027
Adjacent sequences: A090669 A090670 A090671 this_sequence A090673 A090674 A090675
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Dec 18 2003
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