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

%I A246865 #42 Jan 16 2022 11:42:29
%S A246865 1,1,2,7,66,3061,1095266,3906746485,165835140118904,
%T A246865 96867653699340061187,883158060528372369857672080,
%U A246865 140546577721904223563711600192372503
%N A246865 Total number of reduced decompositions for all permutations in S_n.
%C A246865 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
%D A246865 Bridget Eileen Tenner, Enumerating in Coxeter Groups (Survey), Advances in Mathematical Sciences, pp 75-82, Springer 2020.
%H A246865 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000001/">The number of ways to write a permutation as a minimal length product of simple transpositions</a>
%H A246865 M. J. Hay, J. Schiff, N. J. Fisch, <a href="http://arxiv.org/abs/1508.03499">Maximal energy extraction under discrete diffusive exchange</a>, arXiv preprint arXiv:1508.03499, 2015
%H A246865 M. J. Hay, J. Schiff, N. J. Fisch, <a href="http://arxiv.org/abs/1604.08573">Available free energy under local phase space diffusion</a>, arXiv preprint arXiv:1604.08573, 2016
%H A246865 M. J. Hay, J. Schiff, N. J. Fisch, <a href="http://w3.pppl.gov/~fisch/fischpapers/2017/Hay_PhysicaA2017.pdf">On extreme points of the diffusion polytope</a>, Physica A 473 (2017) 225-236. <a href="http://dx.doi.org/10.1016/j.physa.2017.01.038">doi:10.1016/j.physa.2017.01.038</a>
%H A246865 R. P. Stanley, <a href="http://dx.doi.org/10.1016/S0195-6698(84)80039-6">On the number of reduced decompositions of elements of Coxeter groups</a>, European J. Combin., 5 (1984), 359-372.
%e A246865 a(4) = 66 is summarized in a table of multiplicity versus length:
%e A246865 length     =   0  1  2  3  4  5  6
%e A246865 multiplicity +---------------------+
%e A246865        1     | 1  3  4  2  .  .  . | = 10   1*10 = 10
%e A246865        2     | .  .  1  4  1  .  . | =  6   2*6  = 12
%e A246865        3     | .  .  .  .  4  .  . | =  4   3*4  = 12
%e A246865        5     | .  .  .  .  .  2  . | =  2   5*2  = 10
%e A246865        6     | .  .  .  .  .  1  . | =  1   6*1  =  6
%e A246865       16     | .  .  .  .  .  .  1 | =  1  16*1  = 16
%e A246865              +---------------------+   --          --
%e A246865                1  3  5  6  5  3  1   = 24   a(4) = 66.
%e A246865 - _Michael Somos_, Sep 07 2014
%Y A246865 Cf. A005118, A008302.
%Y A246865 Row sums of A289778.
%K A246865 nonn,more
%O A246865 0,3
%A A246865 _Sara Billey_, Sep 05 2014
%E A246865 a(0)=1 prepended and a(7)-a(11) from _Alois P. Heinz_, Jul 10 2017