A quota system is a voting rule for two alternatives in which each voter favors one alternative or records a tie. The rule depends only on the numbers of votes of each kind, treats the two alternatives symmetrically, and cannot become less favorable to an alternative when a voter changes from opposing it to a tie or from a tie to favoring it (Taylor 1995; Llamazares 2014, pp. 112-113).
For 
voters, where 
is a nonnegative integer,
a canonical list of integer
quotas 
satisfies 
,
for 
, and 
for 
. If exactly 
voters record ties, an alternative is chosen iff
it receives at least 
votes. If neither alternative meets this quota, the result is a tie. The bound 
prevents both alternatives
from meeting the quota, and 
represents a tie for every profile with exactly 
tie votes.
Let 
be the number of quota systems for
voters and 
the number of quota systems for which 
, so
 |
(1)
|
where 
is the floor function (Young et al. 1995;
Llamazares 2014, p. 116). For 
, 1, 2, ..., this produces the sequence
1, 2, 3, 6, 10, 20, 35, 70, 126, ... (OEIS A001405,
at index 
).
These are middle binomial coefficients in
successive rows of Pascal's triangle, starting
with row 1. The function 
may be defined recursively by 
and
 |
(2)
|
where 
is a nonnegative integer and 
is a Catalan number (Young
et al. 1995). The function 
satisfies
 |
(3)
|
for integers 
and 
(Young et al. 1995). Set 
for integer 
outside this range. On its stated domain, 
satisfies the quota rule.
