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A246865 Total number of reduced decompositions for all permutations in S_n. 2
1, 1, 2, 7, 66, 3061, 1095266, 3906746485, 165835140118904, 96867653699340061187, 883158060528372369857672080, 140546577721904223563711600192372503 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..11.

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

Cf. A005118, A008302.

Row sums of A289778.

Sequence in context: A116985 A042051 A196925 * A133237 A099660 A341088

Adjacent sequences:  A246862 A246863 A246864 * A246866 A246867 A246868

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

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Last modified August 8 09:40 EDT 2022. Contains 356009 sequences. (Running on oeis4.)