OFFSET
0,3
COMMENTS
A decomposition of a permutation is a product of adjacent transpositions. A reduced decomposition is one of minimal length which is also the number of inversions of the permutation and there may be more than one reduced decomposition. The largest number (multiplicity) of reduced decompositions of a permutation in S_n is A005118(n) for the permutation which reverses the order of all elements and all of its reduced decompositions have length n(n-1)/2 which is the maximum number of inversions. - Michael Somos, Sep 07 2014
REFERENCES
Bridget Eileen Tenner, Enumerating in Coxeter Groups (Survey), Advances in Mathematical Sciences, pp 75-82, Springer 2020.
LINKS
FindStat - Combinatorial Statistic Finder, The number of ways to write a permutation as a minimal length product of simple transpositions
M. J. Hay, J. Schiff, N. J. Fisch, Maximal energy extraction under discrete diffusive exchange, arXiv preprint arXiv:1508.03499, 2015
M. J. Hay, J. Schiff, N. J. Fisch, Available free energy under local phase space diffusion, arXiv preprint arXiv:1604.08573, 2016
M. J. Hay, J. Schiff, N. J. Fisch, On extreme points of the diffusion polytope, Physica A 473 (2017) 225-236. doi:10.1016/j.physa.2017.01.038
R. P. Stanley, On the number of reduced decompositions of elements of Coxeter groups, European J. Combin., 5 (1984), 359-372.
EXAMPLE
a(4) = 66 is summarized in a table of multiplicity versus length:
length = 0 1 2 3 4 5 6
multiplicity +---------------------+
1 | 1 3 4 2 . . . | = 10 1*10 = 10
2 | . . 1 4 1 . . | = 6 2*6 = 12
3 | . . . . 4 . . | = 4 3*4 = 12
5 | . . . . . 2 . | = 2 5*2 = 10
6 | . . . . . 1 . | = 1 6*1 = 6
16 | . . . . . . 1 | = 1 16*1 = 16
+---------------------+ -- --
1 3 5 6 5 3 1 = 24 a(4) = 66.
- Michael Somos, Sep 07 2014
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Sara Billey, Sep 05 2014
EXTENSIONS
a(0)=1 prepended and a(7)-a(11) from Alois P. Heinz, Jul 10 2017
STATUS
approved