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%I #42 Jan 16 2022 11:42:29
%S 1,1,2,7,66,3061,1095266,3906746485,165835140118904,
%T 96867653699340061187,883158060528372369857672080,
%U 140546577721904223563711600192372503
%N Total number of reduced decompositions for all permutations in S_n.
%C 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 Bridget Eileen Tenner, Enumerating in Coxeter Groups (Survey), Advances in Mathematical Sciences, pp 75-82, Springer 2020.
%H 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 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 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 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 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 a(4) = 66 is summarized in a table of multiplicity versus length:
%e length = 0 1 2 3 4 5 6
%e multiplicity +---------------------+
%e 1 | 1 3 4 2 . . . | = 10 1*10 = 10
%e 2 | . . 1 4 1 . . | = 6 2*6 = 12
%e 3 | . . . . 4 . . | = 4 3*4 = 12
%e 5 | . . . . . 2 . | = 2 5*2 = 10
%e 6 | . . . . . 1 . | = 1 6*1 = 6
%e 16 | . . . . . . 1 | = 1 16*1 = 16
%e +---------------------+ -- --
%e 1 3 5 6 5 3 1 = 24 a(4) = 66.
%e - _Michael Somos_, Sep 07 2014
%Y Cf. A005118, A008302.
%Y Row sums of A289778.
%K nonn,more
%O 0,3
%A _Sara Billey_, Sep 05 2014
%E a(0)=1 prepended and a(7)-a(11) from _Alois P. Heinz_, Jul 10 2017