Let
be a finite of infinite set (of variables) and consider the ring of
polynomials
and the ring of power series
over a
commutative ring
with unit element in the commuting variables from
.
A polynomial or power series
is called
symmetric
if for any two finite sequences of indeterminates
and
from
and any sequence of exponents
,
the
coefficients in
of
and
are the same.
Quasi-symmetric formal power series
are a generalization
introduced
by
I.M. Gessel,
[a2],
in connection with the
combinatorics of plane partitions
and
descent sets of permutations
[a3].
This time one takes a totally ordered set of indeterminates, e.g.
,
with
the ordening that of the natural numbers, and the condition is that
the
coefficients of
and
are equal for all totally ordered sets of
indeterminates
and
.
For example,
is a quasi-symmetric polynomial in three variables that is
not symmetric.
Products and sums of quasi-symmetric polynomials and power
series are
again quasi-symmetric (obviously), and thus one has, for
example, the ring
of quasi-symmetric power series
in countably many commuting variables over the integers and its
subring
of quasi-symmetric polynomials in finite of countably many
indeterminates,
which are the quasi-symmetric power series of bounded degree.
Given a word
over
,
also called a
composition
in this context,
consider the
quasi-monomial function
defined by
.
These form a basis over the integers of
.
The algebra of quasi-symmetric functions is dual to the
Leibniz–Hopf algebra,
or, equivalently to the
Solomon descent algebra,
more
precisely,
to the direct sum
of the Solomon descent algebras
of the symmetric groups (cf. also
Symmetric group),
[a5],
with a new
multiplication over which the direct sum of the original
multiplications is distributive. See
[a1],
[a4].
The algebra of quasi-symmetric functions in countably many
indeterminates over the integers,
,
is a free polynomial algebra over the integers,
[a6].
There is a completely different notion in the theory of functions of
a complex variable that also goes by the name
quasi-symmetric function;
cf., e.g.,
[a7].