Let $m_G$ denote the number of perfect matchings of the graph $G$. We introduce a number of combinatorial tools for determining the parity of $m_G$ and giving a lower bound on the power of 2 dividing $m_G$. In particular, we introduce certain vertex sets called channels, which correspond to elements in the kernel of the adjacency matrix of $G$ modulo $2$. A result of Lovász states that the existence of a nontrivial channel is equivalent to $m_G$ being even. We give a new combinatorial proof of this result and strengthen it by showing that the number of channels gives a lower bound on the power of $2$ dividing $m_G$ when $G$ is planar. We describe a number of local graph operations which preserve the number of channels. We also establish a surprising connection between 2-divisibility of $m_G$ and dynamical systems by showing an equivalency between channels and billiard paths. We exploit this relationship to show that $2^{\frac{\gcd(m+1,n+1)-1}{2}}$ divides the number of domino tilings of the $m\times n$ rectangle. We also use billiard paths to give a fast algorithm for counting channels (and hence determining the parity of the number of domino tilings) in simply connected regions of the square grid.
We adapt the hypercube decompositions introduced by Blundell-Buesing-Davies-Veličković-Williamson to prove the Combinatorial Invariance Conjecture for Kazhdan-Lusztig polynomials in the case of elementary intervals in Sn. This significantly generalizes the main previously-known case of the conjecture, that of lower intervals.
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