A227404 Total number of inversions in all permutations of order n consisting of a single cycle.

0, 0, 1, 4, 22, 140, 1020, 8400, 77280, 786240, 8769600, 106444800, 1397088000, 19718899200, 297859161600, 4794806016000, 81947593728000, 1482030950400000, 28277150533632000, 567677135241216000, 11961768206868480000, 263969867887165440000

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+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

Links

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.

Formula

For n>2, a(n) = n! * (3*n-1)/12. - Vaclav Kotesovec, Feb 14 2014

Example

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.

Mathematica

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

Crossrefs

Cf. A001809, A211606, A216239.

Sequence in context: A181784 A003287 A077056 * A366677 A190271 A045744

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