3 sp

There is a large combinatorial literature on this, with many extensions and applications (including, notably, the MacMahon Master theorem). I am giving below a few pointers so you can start - the real literature is very large.

1) multivariate version. See "Combinatorial Enumeration" by Goulden and Jackson. See also this Gessel's paper with a combinatorial proof (there are other comb. proofs), many references, etc.

2) $q$- and non-commutative extensions. There are several $q$-analogues due to Andrews (see here, Garsia (see here for ref and bijective proof), and Gessel (see here for both $q$- and non-commutative generalization). Finally, see this (more general) non-commutative extension using quasideterminants.

3) the version in the language of species (see also this MO question) - see this paper by Gessel and Labelle.

2 sp

There is a large combinatorial literature on this, with many extensions and applications (including, notably, the MacMahon Master theorem. I am giving below a few pointers so you can start - the real literature is very large.

1) multivariate version. See "Combinatorial Enumeration" by Goulden and Jackson. See also this Gessel's paper with a combinatorial proof (there are other comb. proofs), many references, etc.

2) $q$- and non-commutative extensions. There are several $q$-analogues due to Andrews (see here, Garsia (see here for ref and bijective proof), and Gessel (see here for both $q$- and non-commutative generalization). Finally, see this (more general) non-commutative extension using quasideterminants".

3) the version in the language of species (see also this MO question) - see this paper by Gessel and Labelle.

1

There is a large combinatorial literature on this, with many extensions and applications (including, notably, the MacMahon Master theorem. I am giving below a few pointers so you can start - the real literature is very large.

1) multivariate version. See "Combinatorial Enumeration" by Goulden and Jackson. See also this Gessel's paper with a combinatorial proof (there are other comb. proofs), many references, etc.

2) $q$- and non-commutative extensions. There are several $q$-analogues due to Andrews (see here, Garsia (see here for ref and bijective proof), and Gessel (see here for both $q$- and non-commutative generalization). Finally, see this (more general) non-commutative extension using quasideterminants".

3) the version in the language of species (see also this MO question) - see this paper by Gessel and Labelle.