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A227404
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Total number of inversions in all permutations of order n consisting of a single cycle.
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3
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0, 0, 1, 4, 22, 140, 1020, 8400, 77280, 786240, 8769600, 106444800, 1397088000, 19718899200, 297859161600, 4794806016000, 81947593728000, 1482030950400000, 28277150533632000, 567677135241216000, 11961768206868480000, 263969867887165440000
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OFFSET
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0,4
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COMMENTS
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The formula trivially follows from the observation that every pair of elements i<j forms an inversion in exactly (binomial(n,2)-n+j-i)*(n-3)! single-cycle permutations. - Max Alekseyev, Jan 05 2018
a(n) is the number of ways to partition a (n+1)X(n+1) square, with the upper left hand corner missing, into ribbons of size n, see Alexandersson, Jordan. - Per W. Alexandersson, Jun 02 2020
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..449
Per Alexandersson, Linus Jordan, Enumeration of border-strip decompositions, arXiv:1805.09778 [math.CO], 2018.
Per Alexandersson, Linus Jordan, Enumeration of border-strip decompositions, Journal of Integer Sequences, Vol. 22 (2019), Article 19.4.5.
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FORMULA
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For n>2, a(n) = n! * (3*n-1)/12. - Vaclav Kotesovec, Feb 14 2014
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EXAMPLE
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a(3) = 4 because the cyclic 3-permutations: (1,2,3), (1,3,2) written in one line (sequence) notation: {2,3,1}, {3,1,2} have 2 + 2 = 4 inversions.
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MATHEMATICA
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Table[Total[Map[Inversions, Map[FromCycles, Map[List, Map[Prepend[#, n]&, Permutations[n-1]]]]]], {n, 1, 8}]
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CROSSREFS
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Cf. A001809, A211606, A216239.
Sequence in context: A181784 A003287 A077056 * A190271 A045744 A243626
Adjacent sequences: A227401 A227402 A227403 * A227405 A227406 A227407
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KEYWORD
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nonn
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AUTHOR
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Geoffrey Critzer, Sep 21 2013
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EXTENSIONS
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a(13)-a(15) from Alois P. Heinz, Sep 26 2013
Terms a(16) and beyond from Max Alekseyev, Jan 05 2018
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STATUS
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approved
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