A particle physicist lacing their shoes

TL;DR

The paper tackles a combinatorial counting problem for topologically distinct lace configurations of an $N$-hole-per-side shoe with $\ell$ strings, establishing an explicit formula for the counting coefficients $c^{(N)}_{p_1\dots p_\ell}$ via a tensor-based formulation. It then shows that these coefficients reproduce a Legendre-polynomial structure in spin-$J=N$ exchange amplitudes, connecting a concrete combinatorial problem to angular-momentum physics and yielding a Yukawa-type potential in coordinate space. In addition, the work derives exact perturbative unitarity bounds for an $O(n)$ scalar theory, expressing constraints in terms of $C_Q(n)=(n+2(Q-1))!!/(n-2)!!$ and describing Argand-plane circle regions that bound couplings and energy scales. Together, the results reveal deep links between combinatorics, tensor contractions, Legendre polynomials, and fundamental scattering theory, with implications for the validity of non-renormalizable interactions.

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.