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
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.
Programs
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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:
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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.
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