Powers of 2 in High-Dimensional Lattice Walks

Let $W_d(n)$ be the number of $2n$-step walks in $\mathbb{Z}^d$ which begin and end at the origin. We study the exponent of $2$ in the prime factorisation of this number; i.e., $w_d(n) = \nu_2(W_d(n))$. We show that, for each $d$, there is a relationship between $w_d(n)$ and the number $s_2(n)$ of $1$s in the binary expansion of $n$. For example, $w_d(n) = s_2(n)$ if $d$ is odd and $w_d(n) = 2s_2(n)$ if $\nu_2(d) = 1$; while $w_d(n) \ge 3s_2(n)$ if $\nu_2(d) = 2$. The pattern changes further when $\nu_2(d) \ge 3$. However, for each $d$, we give the best analogous estimate of $w_d(n)$ together with a description of all $n$ where equality is attained. The methods we develop apply to a wider range of problems as well, and so might be of independent interest.