A monomial symmetric polynomial is the sum of all distinct monomials obtained by permuting a fixed list of exponents
among the variables. For a positive
integer 
, let 
be a partition
with 
,
padded with zeros to length 
. If 
is the set of distinct permutations
of this list, then
 |
(1)
|
These polynomials are also called monomial symmetric functions (Dumitriu et al. 2007). Each distinct monomial
occurs once, even when parts of 
are repeated. In particular, the sum is not taken over
all 
permutations with their multiplicities.
For example,
 |  |  |
(2)
|
 |  |  |
(3)
|
For 
,
set 
.
The empty partition gives the constant 1.
For a fixed nonnegative integer 
, the 
with 
a partition of 
into at most 
parts form a vector basis
of the symmetric polynomials that are homogeneous polynomials of total
degree 
over any field. Indeed, symmetry
forces the coefficients to be constant on each group orbit of exponent lists, so every such polynomial
is a linear combination of these sums. Different
sums have disjoint monomial supports, so the sums are
linearly independent.
If 
is the number of parts of 
equal to 
, then the number of distinct monomials
is
 |
(4)
|
for 
,
where the evaluation is over the integers. The denominator
accounts for the 
repeated zeros and for equal positive parts. For example,
.
For a positive integer 
, 
is the 
th power sum, while 
with 
parts is the 
th elementary
symmetric polynomial.
