// Description of sigma(W) and f_W for the Coxeter group:
Coxeter group: Finitely presented group on 3 generators
Relations
    s * t * s = t * s * t
    s * u = u * s
    (t * u)^2 = (u * t)^2
    s^2 = Id($)
    t^2 = Id($)
    u^2 = Id($)


// W has order 48.


// 21 elements of W do not lie in \sigma(W).
// They are:

c := [ 
u * t * u,
t * u * t,
u * t * s * u,
t * u * t * s,
u * t * s * u * t,
t * s * u * t * s,
s * u * t * u,
t * s * u * t * u,
u * t * s * u * t * u,
(t * s * u)^2,
t * u * t * s * u * t * u,
s * t * u * t,
s * t * s * u * t,
(s * u * t)^2,
s * u * t * s * u * t * u,
t * s * u * t * s * u * t,
t * s * u * t * s * u * t * u,
s * t * u * t * s,
s * t * s * u * t * s,
s * t * u * t * s * u * t * u,
s * t * s * u * t * s * u * t 
 ];


 // The b-basis on the elements not in Sigma(W):
 // (H(w) denotes the KL-basis elt corresponding to w.) 

(1)C(u) + (1)C(u * t * u).
(1)C(t) + (1)C(t * u * t).
(1)C(s * u) + (1)C(u * t * s * u).
(1)C(t * s) + (1)C(t * u * t * s).
(1)C(s * u * t) + (1)C(u * t * s * u * t).
(1)C(s * t * s) + (1)C(t * s * u * t * s).
(1)C(s * u) + (1)C(s * u * t * u).
(1)C(t * s * u) + (1)C(t * s * u * t * u).
(1)C((t * u)^2) + (1)C(s * u) + (1)C(u * t * s * u) + (1)C(s * u * t * u) + (1)C(u * t * s * u * t * u).
(1)C(s * t * s * u) + (1)C((t * s * u)^2).
(1)C(t * u * t * s * u) + (1)C(t * u * t * s * u * t * u).
(1)C(s * t) + (1)C(s * t * u * t).
(1)C(s * t * s) + (1)C(s * t * s * u * t).
(1)C(s * u * t * s) + (1)C((s * u * t)^2).
(1)C(s * t * u * t * u) + (1)C(s * u * t * s * u * t * u).
(1)C(s * t * s) + (1)C(t * s * u * t * s) + (1)C(s * t * s * u * t) + (1)C(t * s * u * t * s * u * t).
(1)C((t * u)^2) + (1)C(t * s * u * t * s * u * t * u).
(1)C(s) + (1)C(s * t * s) + (1)C(s * t * u * t * s).
(v^-1 + v)C(s * t * s) + (1)C(s * t * s * u * t * s).
(1)C(s * t * u * t * s * u) + (1)C(s * t * u * t * s * u * t * u).
(1)C(s * t * s * u * t * s) + (1)C(s * t * s * u * t * s * u * t).