-multinomial coefficient
Let 
be the multiset that has 
copies of 
, 
. 
denotes the set of permutations
of 
for all distinct orderings of the 
integers.
The number of elements in is the multinomial coefficient
(§26.4) 
.
Additional information can be found in Andrews (1976, pp. 39–45).
The definitions of inversion number and major index can be extended to
permutations of a multiset such as
. Thus
, and
The -multinomial coefficient is defined in terms of Gaussian
polynomials (§26.9(ii)) by
![\genfrac{[}{]}{0.0pt}{}{a_{1}+a_{2}+\cdots+a_{n}}{a_{1},a_{2},\ldots,a_{n}}_{{%
q}}=\prod_{{k=1}}^{{n-1}}\genfrac{[}{]}{0.0pt}{}{a_{k}+a_{{k+1}}+\cdots+a_{n}}%
{a_{k}}_{{q}},](./26/16/E1.png)
![\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}](./26/16/m12.png){kind=small}
: -binomial coefficient (or Gaussian polynomial),
: nonnegative integer,
: nonnegative integer,
: numbers and
: parameter
and again with we have
![\sum_{{\sigma\in\mathop{\mathfrak{S}_{{S}}\/}\nolimits}}q^{{\mathop{\mathrm{%
inv}}(\sigma)}}=\genfrac{[}{]}{0.0pt}{}{a_{1}+a_{2}+\cdots+a_{n}}{a_{1},a_{2},%
\ldots,a_{n}}_{{q}},](./26/16/E2.png)
: element of,
: -binomial coefficient (or Gaussian polynomial),
: set of permutations of 
,
: nonnegative integer,
: numbers,
: multiset and
: parameter
![\sum_{{\sigma\in\mathop{\mathfrak{S}_{{S}}\/}\nolimits}}q^{{\mathop{\mathrm{%
maj}}(\sigma)}}=\genfrac{[}{]}{0.0pt}{}{a_{1}+a_{2}+\cdots+a_{n}}{a_{1},a_{2},%
\ldots,a_{n}}_{{q}}.](./26/16/E3.png)
: element of,
: -binomial coefficient (or Gaussian polynomial),
: set of permutations of ,
: nonnegative integer,
: numbers,
: multiset and
: parameter
