A particle physicist lacing their shoes

Abstract This letter presents a counting problem and its solution in two forms, one derived by the author and one worked out by Young (private correspondence). The problem can be mapped to both (i) ways to lace a N -holes-per-side shoe with $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> shoestrings and (ii) a sum over the indexes of 2 N -tensors made of symmetric 2-tensors. The coefficients that answer this question are then connected to spin $$J=N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>J</mml:mi> <mml:mo>=</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> boson exchange amplitudes and perturbative unitarity constraints, deriving relations with Legendre polynomials and sum rules.