

A227404


Total number of inversions in all permutations of order n consisting of a single cycle.


3



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

0,4


COMMENTS

The formula trivially follows from the observation that every pair of elements i<j forms an inversion in exactly (binomial(n,2)n+ji)*(n3)! singlecycle 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


LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..449
Per Alexandersson, Linus Jordan, Enumeration of borderstrip decompositions, arXiv:1805.09778 [math.CO], 2018.
Per Alexandersson, Linus Jordan, Enumeration of borderstrip decompositions, Journal of Integer Sequences, Vol. 22 (2019), Article 19.4.5.


FORMULA

For n>2, a(n) = n! * (3*n1)/12.  Vaclav Kotesovec, Feb 14 2014


EXAMPLE

a(3) = 4 because the cyclic 3permutations: (1,2,3), (1,3,2) written in one line (sequence) notation: {2,3,1}, {3,1,2} have 2 + 2 = 4 inversions.


MATHEMATICA

Table[Total[Map[Inversions, Map[FromCycles, Map[List, Map[Prepend[#, n]&, Permutations[n1]]]]]], {n, 1, 8}]


CROSSREFS

Cf. A001809, A211606, A216239.
Sequence in context: A181784 A003287 A077056 * A190271 A045744 A243626
Adjacent sequences: A227401 A227402 A227403 * A227405 A227406 A227407


KEYWORD

nonn


AUTHOR

Geoffrey Critzer, Sep 21 2013


EXTENSIONS

a(13)a(15) from Alois P. Heinz, Sep 26 2013
Terms a(16) and beyond from Max Alekseyev, Jan 05 2018


STATUS

approved



