  In 2004, Rosas and Sagan developed the theory of symmetric functions in
noncommuting variables, achieving results analogous to classical symmetric
functions. On the other hand, in 2004, Desrosiers, Lapointe and Mathieu
introduced the theory of symmetric functions in superspace, which involve both
commuting and anticommuting variables, extending the classic theory. Here, we
introduce symmetric functions in noncommuting variables in superspace. We
define the classical symmetric functions in noncommuting variables to
superspace: monomials, power sums, elementaries and complete homogeneous, which
generalize both the ones studied by Rosas and Sagan and the ones studied by
Desrosiers, Lapointe and Mathieu. We also define Schur--type functions in
noncommuting variables in superspace.

Combinatorics