What are the analogues of Littlewood-Richardson coefficients for monomial symmetric polynomials? The product of monomial symmetric polynomials can be expressed as
$m_{\lambda} m_{\mu} = \Sigma c_{\lambda\mu}^{\nu}m_{\nu}$
for some constants $c_{\lambda\mu}^{\nu}$.
In the case of Schur polynomials, these constants are called the Littlewood-Richardson coefficients. What are they called for monomial symmetric polynomials, and how do I calculate them?
 A: 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.
A: It is a matter of basic linear algebra to get the answer. Let me
reformulate the question.

*

*Let $s_{\cdot}$ be the Schur polynomials and $c_{i,j}^k$ be
the Littlewood-Richardson coefficients. Namely, with Einstein
summation convention, $$s_{i}s_{j} = c_{i,j}^{k} s_{k}$$


*Let $m_{\cdot}$ be the symmetric monomials and $d_{a,b}^c$ be
such that $$m_{a}m_{b} = d_{a,b}^{c} m_{c}.$$
We want an expression of the $d$'s in terms of the $c$'s.

Deduction
It is wellknown that both $s$ and $m$ forms a basis for the ring
of symmetric function over $\mathbb{Q}$. So there are matrices
$P, Q$ inverse to each other such that $m_a = P_a^i s_i$ and $s_i
= Q_i^a m_a$ for any $a$ and $i$.
Then $m_a m_b$ equals $(P_a^i s_i)(P_b^j s_j)$, so
$$P_a^i P_b^j c_{i,j}^k = d_{a,b}^c P_c^k.$$
Since $Q$ is the inverse of $P$, we have the formula
$$d_{a,b}^c = P_a^i P_b^j Q_k^c c_{i,j}^k.$$
