A109281 Triangle T(n,k) of elements of n-th Weyl group of type B whose reduced word uses n-k generators.

1, 1, 1, 5, 2, 1, 35, 9, 3, 1, 309, 56, 14, 4, 1, 3287, 443, 84, 20, 5, 1, 41005, 4298, 623, 120, 27, 6, 1, 588487, 49937, 5629, 859, 165, 35, 7, 1, 9571125, 680700, 61300, 7360, 1162, 220, 44, 8, 1, 174230863, 10683103, 793402, 75714, 9584, 1544, 286, 54, 9, 1

Offset: 0

Author

Mike Zabrocki, Aug 19 2005

Keywords

• nonn
• tabl

Comments

Row sums are 2^n n!.

G.f. for k-th column is given by (1-1/g(x))^(k-1)*g(2x)/g(x).

Examples

			T(3,1)=9 because B_3 is generated by {t,s1,s2} where t^2=s1^2=s2^2=(s1 s2)^3=(t s1)^4=(t s2)^2=1.
The 9 elements which only use 2 generators are {s1 s2, s1 s2 s1, s2 s1, s2 t, t s1, s1 t s1, s1 t s1 t, s1 t, t s1 t}.
Triangle starts:
1;
1, 1;
5, 2, 1;
35, 9, 3, 1;
309, 56, 14, 4, 1;
...
		

Links

• N. Bergeron, C. Hohlweg, M. Zabrocki, Posets related to the connectivity set of Coxeter groups, arXiv:math/0509271 [math.CO], 2005-2006.

Crossrefs

Cf. A109253, A003319, A085771, A059438.

For the similar sequence in type D, see A112226.

Programs

Maple

f:=proc(n,k) local gx; gx:=add(i!*x^i,i=0..n); coeff(series((1-1/gx)^k*subs(x=2*x,gx)/gx,x,n+1),x,n); end:

Mathematica

nmax = 9;
g[x_] = Sum[n!*x^n, {n, 0, nmax}];
gf[x_, t_] = g[2*x]/(t + (1 - t)*g[x]);
T[n_, k_] := SeriesCoefficient[gf[x, t], {x, 0, n}] // SeriesCoefficient[#, {t, 0, k}]&;
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 25 2017 *)

Formula

G.f.: g(2x)/(t+(1-t)g(x)) where g(x) = sum_{n>=0} n! x^n.