If his partner is absent from work, is it permissible for him to deduct from his earnings for that reason?

Firstly: 

The basic
principle in the case of a dispute between two partners or parties to an
agreement is that they should refer to the conditions agreed upon in the
contract between them, so long as these conditions are not contrary to
Islamic teaching, because Allah, may He be exalted, says (interpretation of
the meaning): “O
you who believe! Fulfill (your) obligations”
[al-Maa’idah 5:1].
And the Prophet (blessings and peace of Allah be upon him) said: “The
Muslims are bound by their conditions, except a condition that forbids
something permissible or permits something forbidden.”

Narrated by
at-Tirmidhi (1352) and Abu Dawood (3594); classed as saheeh by al-Albaani in
Saheeh at-Tirmidhi. 

Based on that, if
there was an agreement between you to deduct from the earnings in the event
of one of the two partners being absent, then there is no doubt that this
deduction is permissible. 

Similarly, if the
normal practice in companies like yours is that if one of the partners is
absent, a deduction is made in that case, and this is well known to you and
your partner, then there is nothing wrong with this deduction, because the
basic fiqhi principle is that that which is well known according to custom
is like a condition that has been stipulated. 

See: Ghamz
‘Uyoon al-Basaa’ir fi Sharh al-Ashbaah wa’n-Nazaa’ir, 4/206 

Secondly: 

If there was no
agreement in writing concerning such situations, and there is no such
consistent practice in the culture of similar companies, then the case of
the partner who was absent must be one of two possibilities: 

1.

That he was absent
for a valid reason, such as sickness or some other problem. In that case no
deduction should be made from his earnings, but you are within your rights
to require him to appoint a deputy, at his own expense, to do his work,
without expecting the company to bear the cost of paying that deputy. If he
refuses to do so, then you are within your rights to cancel the partnership,
because the agreement was to work together, and if he does not adhere to
that, then you have the right to cancel. 

2.

That his absence
was not for a valid reason; rather it was due to carelessness or a reason
that he could have overcome. In this case there was a difference of opinion
among the scholars – may Allah have mercy on them – as to whether the
partner is entitled to his earnings in full, or not. 

Some of the
scholars are of the view that the partner is entitled to his full share of
the earnings, according to what is agreed upon in the contract, as it is
sufficient that the other partner has the right to cancel the contract if
any of the conditions are not met. If he did not decide to cancel it before
the earnings were known, then the earnings are to be shared between them in
accordance with what they agreed upon. 

It says in
Majallat al-Ahkaam al-‘Adliyyah: Item no. 1349:

Entitlement to a
share of the earnings is based on the conditions stipulated in the
partnership contract, not on the work that was done. Based on that, if the
partner whose stipulated role was to work did not work as stipulated, he is
still to be regarded as if he did do the work. For example, if it was
stipulated that both partners in a valid partnership should work, and only
one of them did the work and the other did not, with or without a valid
excuse, then because they are deputies of one another, if one partner does
the work it is as if the other partner did it, therefore the earnings are to
be shared between them in to the manner stipulated. 

Ibn Qudaamah (may
Allah have mercy on him) said: 

If one of them
works but the other does not, the earnings are to be shared between them,
whether (the latter) did not work because of sickness or otherwise. If one
of them asked the other to work with him, or to appoint someone else in his
stead, he is within his rights to do so. If (the latter) refuses, then (the
former) is within his rights to cancel the agreement.

End quote from
al-Mughni, 5/7 

The second view
concerning this issue is that the partner (who does not do his work) does
not deserve his full share of the earnings because he has not upheld his
part of the agreement. 

Al-Mirdaawi (may
Allah have mercy on him) said: 

What is understood
from the words “if one of them is sick, the earnings are for both of them”
is that if he did not do the work for no good reason then the earnings are
not to be shared between them. This is one of the two views [of the
Hanbalis], and the author [i.e., Ibn Qudaamah] thinks that this is the
correct view. 

End quote from
al-Insaaf, 5/461 

Shaykh Ibn
‘Uthaymeen (may Allah have mercy on him) said: 

The more correct
view concerning this issue is that if he did not do the work for no valid
reason, then he does not deserve the earnings of the time during which he
did not work, with no valid excuse. End quote from ash-Sharh al-Mumti‘,
9/436 

Whatever the case,
whether his absence was for a valid reason or not, you are within your
rights to cancel the previous agreement and make a new agreement in which
you state how earnings are to be shared out if either of the two partners is
absent. This is more prudent for both of you, and less likely to lead to
conflicts or create doubts concerning the honesty of either partner. 

And Allah knows
best.