A particle physicist lacing their shoes

Abstract

This letter presents the solution to a counting problem, to the best of our knowledge not known in full generality, which can be mapped to both (i) ways to lace a $N$-holes-per-side shoe with $\ell$ shoestrings and (ii) a sum over the indexes of $2N$-tensors made of symmetric 2-tensors. The coefficients that answer this question are then connected to spin $J=N$ boson exchange amplitudes and perturbative unitarity constraints, deriving relations with Legendre polynomials and sum rules.