What are the analogues of Littlewood-Richardson coefficients for monomial symmetric polynomials?

I found this reference, where the authors deal with the products you asked for.

EDIT The reference is

A MAPLE program for calculations with Schur functions by M.J. Carvalho, S. D’Agostino Computer Physics Communications 141 (2001) 282–295

From the paper (p.5 chap. 3.1 Multiplication and division of $m$-functions):

Let’s define the result of the addition and subtraction of two partitions $(\mu_1,\mu_2, . . .)$ and $(\nu_1, \nu_2, . . .)$ as being the partition whose parts are $(\mu_1 ± \nu_1,\mu_2 ± \nu_2, . . .)$. For these operations to be meaningful, it is necessary that both partitions have an equal number of parts; if they do not, then one increases the number of parts of the shortest one by adding enough zeros at the end. ... The multiplication (and division) of two m-functions are then defined as $$ m_{\alpha} m_{\beta} = \Sigma I_{\gamma}m_{\gamma} $$ and $$ m_{\alpha}/ m_{\beta} = \Sigma I_{\gamma'}m_{\gamma'} $$ where the partitions $\gamma$,$\gamma'$ result from adding to or subtracting, respectively, from $\alpha$ all distinct partitions obtained by permuting in all possible ways the parts of $\beta$. Clearly, all $m$-functions involved are functions of the same $r$ indeterminates, i.e. have the same number of total parts. The coefficient $I_\nu$, with $\nu = \gamma$ is given by $$ I_\nu=n_\nu \frac{\dim (m_\alpha)}{\dim (m_\nu)} $$ where $n_\nu$ is the number of times the same partition $\nu$ appears in the process of adding or subtracting partitions referred to above.

As far as I read, they don't give a special name to these coefficients.