A246865 Total number of reduced decompositions for all permutations in S_n.

1, 1, 2, 7, 66, 3061, 1095266, 3906746485, 165835140118904, 96867653699340061187, 883158060528372369857672080, 140546577721904223563711600192372503

Offset: 0

Author

Sara Billey, Sep 05 2014

Keywords

• nonn
• more

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

Examples

			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
		

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.

Crossrefs

Cf. A005118, A008302.

Row sums of A289778.

Extensions

a(0)=1 prepended and a(7)-a(11) from Alois P. Heinz, Jul 10 2017