, which is the coefficient of the Schur function 
in the product 
. combinat::lrcalc – Interface with lrcalc
combinat::lrcalc is an interface with Anders Buch's Littlewood-Richardson Calculator.
Details:
combinat::lrcalc is a dynamic module interface with Anders Buch's C library for computing Littlewood-Richardson coefficients and the related operations on Schur symmetric functions. See http://home.imf.au.dk/abuch/lrcalc/ for details on this library.
This library is designed as a raw low-level interface to lrcalc, and is not necessarily consistent with the rest of the combinat library.
See e.g. examples::SymmetricFunctions for higher-level interfaces.
lrcoef – Littlewood-Richardson coefficient
combinat::lrcalc::lrcoef(partition nu, partition lambda, partition mu)
Returns the Littlewood-Richardson coefficient , which is the coefficient of the Schur function in the product .
skew – Expansion of a skew Schur function in the basis of Schur functions
combinat::lrcalc::skew(partition nu, partition lambda)
Returns the list of all partitions 
for which the Littlewood-Richardson coefficient 
is non-zero, with that coefficient in front.
mult – Expansion of a product of two Schur functions in the basis of Schur functions
combinat::lrcalc::mult(partition lambda, partition mu)
Returns the list of all partitions 
for which the Littlewood-Richardson coefficient 
is non-zero, with that coefficient in front.
coprod – Expansion of the coproduct of a Schur function
combinat::lrcalc::coprod(partition nu)
Returns the list of all pairs of partitions 
and 
for which the Littlewood-Richardson coefficient 
is non-zero, with that coefficient in front.
We start by computing the Littlewood-Richardson coefficient for 
, 
and 
:
combinat::lrcalc::lrcoef([3, 2, 1], [2, 1], [2, 1])
Most of the remaining methods return linear combination of partitions (or of pairs of partitions), which are represented in the usual poly2list format. For example, the skew Schur function 
expands on Schur functions as 
:
combinat::lrcalc::skew([3, 2, 1], [2, 1])
This gives the expansion of the product of the two Schur functions
combinat::lrcalc::mult([2, 1], [2, 1])
Coproduct are also computed by
combinat::lrcalc::coprod([3, 2, 1])
And finally if one only needs one coefficient
combinat::lrcalc::lrcoef([12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1],
[8, 7, 6, 5, 4, 3, 2, 1],
[8, 7, 6, 6, 5, 4, 3, 2, 1])
Changes in MuPAD 3.1
New Function.