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.

0, 0, 1, 1, 2, 1, 13, 7, 3, 1, 135, 40, 12, 4, 1, 1537, 293, 66, 18, 5, 1, 19811, 2646, 451, 100, 25, 6, 1, 289073, 28887, 3753, 663, 143, 33, 7, 1, 4741923, 374820, 37798, 5232, 940, 196, 42, 8, 1, 86705417, 5676121, 457508, 49444, 7174, 1294, 260, 52, 9, 1

Offset: 0

Author

Mike Zabrocki, Aug 28 2005

Keywords

• nonn
• tabl

Comments

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)

Examples

			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.
		

Links

• N. Bergeron, C. Hohlweg, M. Zabrocki, Posets related to the connectivity set of Coxeter groups

Crossrefs

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

Programs

Maple

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;

Mathematica

max = 10;
fA = 1 - 1/Sum[n!*x^n, {n, 0, max}] + O[x]^max;
fD = (3 + Sum[2^n*n!*x^n, {n, 0, max}])/(2*Sum[n!*x^n, {n, 0, max}]) + x - 2 + O[x]^max;
f = (2*t*fA - 2*t*x + t^2*x*fA + fD)/(1 - t*fA);
row[n_] := CoefficientList[ SeriesCoefficient[f, {x, 0, n}], t];
Join[{{0}}, {{0, 1}}, Table[row[n], {n, 2, max - 1}]] // Flatten (* Jean-François Alcover, Nov 28 2017 *)

Formula

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}