# Hooks, monomers, dimers and Young diagrams: Part II

As promised, I've upgraded my last question.

Consider the $$k$$-by-$$n$$ partition $$\lambda_n=(n,\dots,n)$$ and its corresponding Young diagram $$Y_{n,k}$$, which is a $$k\times n$$ rectangle of cells. Now, start tiling $$Y_{n,k}$$ using monomers ($$1\times1$$ squares) and dimers ($$1\times2$$ and $$2\times1$$ rectangles).

Next, insert the hook-lengths $$h(\square)$$ into each cell $$\square\in Y_{n,k}$$. Associate a weight: a monomer at $$\square$$ receives $$h(\square)$$, a dimer sitting on $$\square$$ and $$\square'$$ gets the product $$h(\square)\cdot h(\square')$$. Each tiling $$T$$ will have weight assigned as the sum of the weights of its monomers and dimers. Let $$b_{n,k}$$ be the entire sum of the weights of all possible tiltings of $$Y_{n,k}$$. For example, if $$n=3, k=1$$ then we get $$b_{3,1}=(3+2+1)+(3\cdot2+1)+(3+2\cdot1)=6+7+5=18.$$ The first few values are: $$b_1=1, b_2=5, b_3=18, b_4=59, b_5=162$$.

QUESTION. Are these generating function $$G_k(x)=\sum_nb_{n,k}x^n$$ rational functions? Or, can you verify this for small $$k$$, such as $$k=2,3,4$$, etc?

Remark. Fedor's comment led to $$G_1(x)=\frac{x(1+x+5x^3-3x^4)}{(1-x-x^2)^4}$$. A rational function.