Origin , remarks and Motivation from Mathematical Physics
1) The problem was proposed by Horst in a recent Oberwolfach's meeting as a combinatorial problem that arises (as a toy problem) from mathematical physics. I dont remember the precise
The context from physicsof this question is explained in Section 4 of J.Feldman, but I will add more information about it when I will have H.Kn\"orrer, E.Trubowitz:"Construction of a 2-d Fermi Liquid", Proc. XIV. International Congress on Mathematical Physics. Editor: Jean Claude Zambrini. World Scientific 2005
"In this section, we formulate an opportunityelementary question about permutations that may be connected with implementing the Pauli exclusion principle in momentum space."
The problem and some variations are directly related to "cancellations betweenFermionic diagrams".
The wider picture (See the Eleven Papers by J.Feldman, H.Knörrer, E.Trubowitz) is toward mathematical understanding and formalism for highly successful physics quantum theories. (In a very very wide sense this is related to Clay's problem on Yang-Mills and Mass gap.)