Entropy of fully packed hard rigid rods on $d$-dimensional hyper-cubic lattices

We determine the asymptotic behavior of the entropy of full coverings of a $L \times M$squarelatticebyrodsofsize$k\times 1$and$1\times k$, in the limit of large $k$. We show that full coverage is possible only if at least one of $L$ and $M$ is a multiple of $k$, and that all allowed configurations can be reached from a standard configuration of all rods being parallel, using only basic flip moves that replace a $k\times k$ square of parallel horizontal rods by vertical rods, and vice versa. In the limit of large $k$, we show that the entropy per site $S_2(k)$ tends to $A{k}^{-2}\ln k$, with $A=1$. We conjecture, based on a perturbative series expansion, that this large-$k$ behavior of entropy per site is super-universal and continues to hold on all $d$-dimensional hyper-cubic lattices, with $d\ge 2$.