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

Following Richard Stanley's pointers regarding my earlier MO question, I decided to "scale-down" the problem and add a slight "twist" to it.

Consider the one-line partition $$\lambda_n=(n)$$ and its corresponding Young diagram $$Y_n$$, which is a $$1\times n$$ straight bar of $$n$$ cells. Now, start tiling $$Y_n$$ using monomers ($$1\times1$$ squares) and dimers ($$1\times2$$ rectangles). There are $$F_{n+1}$$ (Fibonacci number) different ways of doing so.

Next, insert the hook-lengths $$h(\square)$$ into each cell $$\square\in Y_n$$. 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 $$b_n$$ be the entire sum of the weights of all possible tiltings of $$Y_n$$. For example, if $$n=3$$ then we get $$b_3=(3+2+1)+(3\cdot2+1)+(3+2\cdot1)=6+7+5=18.$$ The first few values are: $$b_1=1, b_2=5, b_3=18, b_4=59, b_5=162$$, this is not listed on OEIS.

QUESTION. If $$G(x)=\sum_nb_nx^n$$ is a generating function of $$b_n$$, is $$G(x)$$ a rational function?

Remark. I might upgrade the question later depending on responses.

POSTSCRIPT. Fedor's comment leads to $$G(x)=\frac{x(1+x+5x^3-3x^4)}{(1-x-x^2)^4}$$.

• it is rational, since $b_{n}=nF_n+b_{n-1}+n(n-1)F_{n-1}+b_{n-2}$ (look at the first cell). – Fedor Petrov Apr 6 at 18:08