A free idempotent monoid is a monoid that satisfies the identity
and is generated by a set of elements. If the generating set of such a monoid is
finite, then so is the free idempotent monoid itself. The number of elements in the
monoid depends on the size of the generating set, and the size the generating set
uniquely determines a free idempotent monoid. On zero letters, the free idempotent
monoid has one element (the identity). With one letter, the free idempotent monoid
has two elements .
With two letters, it has seven elements: . In general, the numbers of elements in
the free idempotent monoids on letters are 1, 2, 7, 160, 332381, ... (OEIS A005345).
These are given by the analytic expression

where
is a binomial coefficient. The product can
be done analytically, giving the sum

in terms of derivatives of the polylogarithm with respect to its index and
the Lerch transcendent with respect to its second argument.

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which ."
Math. Proc. Cambridge Philos. Soc.48, 35-40, 1952.Lallement,
G. Semigroups
and Combinatorial Applications. New York: Wiley, 1979.