> There is something *extremely* fishy about the base ring here!  It's
> a *multivariate* polynomial ring:

Here is the culprit:

sage: type(PolynomialRing(ZZ, 1, 'x'))
<class 
'sage.rings.polynomial.multi_polynomial_ring.MPolynomialRing_polydict_domain'>

while

sage: type(PolynomialRing(ZZ, 'x'))
<class 'sage.rings.polynomial.polynomial_ring.PolynomialRing_integral_domain'>

I've created a ticket: http://www.sagetrac.org/sage_trac/ticket/764
and will post a patch later on.

> sage: s = hlqp5_m.change_ring(ZZ['x'])
> sage: time w=s^(-1)
> CPU times: user 0.10 s, sys: 0.00 s, total: 0.10 s
>
> Even 8 works pretty quickly:
>
> sage: hlqp8 = [ symmetrica.hall_littlewood(p) for p in Partitions(8) ]
> sage: hlqp8_m = matrix([[ x.coefficient(p) for p in Partitions(8) ]
> for x in hlqp8])
> sage: s = hlqp8_m.change_ring(ZZ['x']); s
> 22 x 22 dense matrix over Univariate Polynomial Ring in x over Integer Ring
> sage: time w=s^(-1)
> CPU times: user 2.13 s, sys: 0.04 s, total: 2.17 s
> Wall time: 2.29
> sage: time t=s*s
> CPU times: user 0.24 s, sys: 0.00 s, total: 0.24 s

Those numbers seem much better.

--Mike

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