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A112226 Table T(n,k) of number of elements of Weyl group of type D of order 2^{n-1} n! such that a reduced word uses exactly n-k distinct simple reflections 0 <= k <= n, n>=1. 1

%I

%S 0,0,1,1,2,1,13,7,3,1,135,40,12,4,1,1537,293,66,18,5,1,19811,2646,451,

%T 100,25,6,1,289073,28887,3753,663,143,33,7,1,4741923,374820,37798,

%U 5232,940,196,42,8,1,86705417,5676121,457508,49444,7174,1294,260,52,9,1

%N Table T(n,k) of number of elements of Weyl group of type D of order 2^{n-1} n! such that a reduced word uses exactly n-k distinct simple reflections 0 <= k <= n, n>=1.

%C The first two rows of this table are not well-defined. This is an analog of the notion of permutations with k components for type D (see A059438)

%H N. Bergeron, C. Hohlweg, M. Zabrocki, <a href="http://arXiv.org/abs/math.CO/0509271">Posets related to the connectivity set of Coxeter groups</a>

%F G.f.: (g(2x) - (2 t x - 4 t - 2 x + 4) g(x) - 4 t + 3)/(2(t + (1-t) g(x))) where g(x) = sum_{n >= 0} n! x^n o.g.f. for first column given by (g(2x)+3)/(2g(x)) + x - 2 o.g.f. for k^th (k>1) column given by ((g(2x)+3)/(2g(x)) + x)*(1-1/g(x))^{k-1} - x (1-1/g(x))^{k-2}

%e D_3 is generated by {s_0,s_1,s_2} where s_0^2 = s_1^2 = s_2^2 = (s_0 s_1)^2 = (s_0 s_2)^3 = (s_1 s_2)^2, the elements of this group can be broken up into 4 sets with reduced words as {1}, {s_0, s_1, s_2}, {s_0 s_1, s_1 s_2, s_2 s_1, s_1 s_2 s_1, s_0 s_2, s_2 s_0, s_0 s_2 s_0} hence T(3,3)=1, T(3,2)=3 and T(3,1)=7. T(3,0)=13 since the remaining 13 elements will have reduced words where all three simple reflections appear.

%p f2:=proc(n,k) local i,gx,g2x; gx:=add(i!*x^i, i=0..n); g2x:=subs(x=2*x,gx); coeff(series(((g2x+3)/(2*gx) + x)*(1-1/gx)^k - x*(1-1/gx)^(k-1),x,n+1),x,n); end: f1:=n->coeff(series((add(2^k*k!*x^k,k=1..n)+4)/add(2*k!*x^k,k=0..n)+x-2,x,n+1),x,n); T:=(n,k)->if k=0 then f1(n) else f2(n,k) fi;

%t max = 10;

%t fA = 1 - 1/Sum[n!*x^n, {n, 0, max}] + O[x]^max;

%t fD = (3 + Sum[2^n*n!*x^n, {n, 0, max}])/(2*Sum[n!*x^n, {n, 0, max}]) + x - 2 + O[x]^max;

%t f = (2*t*fA - 2*t*x + t^2*x*fA + fD)/(1 - t*fA);

%t row[n_] := CoefficientList[ SeriesCoefficient[f, {x, 0, n}], t];

%t Join[{{0}}, {{0, 1}}, Table[row[n], {n, 2, max - 1}]] // Flatten (* _Jean-Fran├žois Alcover_, Nov 28 2017 *)

%Y Cf. A112225, A003319, A109253, A109281, A085771, A059438.

%K nonn,tabl

%O 0,5

%A _Mike Zabrocki_, Aug 28 2005

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Last modified May 7 19:23 EDT 2021. Contains 343652 sequences. (Running on oeis4.)