A useful object when dealing with domino tableau are the LLT Polynomials. They are symmetric functions and are defined as follows. For a skew shape and parameter , the corresponding LLT polynomial is:
If the content of P has 1s, 2s, ... , ns, then and we define spin(P) to be half the number of vertical dominoes in P
Calculating an LLT Polynomial
We wish to calculate the LLT polynomial associated with the shape [3,3], given variables . Because there are two variables, we look at all legal domino tableaux that have shape [3,3] that do not contain any numbers other than one or two. These can be seen below.
Considering the definition, and the diagram above, we can see that