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

Author

Geoffrey Critzer, Sep 21 2013

Keywords

• nonn

Comments

The formula trivially follows from the observation that every pair of elements iMax 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

Examples

			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.
		

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.

Crossrefs

Cf. A001809, A211606, A216239.

Programs

Mathematica

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

Formula

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

Extensions

a(13)-a(15) from Alois P. Heinz, Sep 26 2013

Terms a(16) and beyond from Max Alekseyev, Jan 05 2018