Minimum permanents on two faces of the polytope of doubly stochastic matrices

We consider the minimum permanents and minimising matrices on the faces of the polytope of doubly stochastic matrices whose nonzero entries coincide with those of, respectively,

U_m,n =

[I_n J_n,m]

[J_m,n 0_m ]

and

V_m,n=

[I_n J_n,m]

[J_m,n J_m,m]

We conjecture that V_m,n is cohesive but not barycentric for 1 < n < m + √m and that it is not cohesive for n ≥ m + √m. We prove that it is cohesive for 1 < n < m + √m and not cohesive for n ≥ 2m and confirm the conjecture computationally for n < 2m ≤ 200. We also show that U_m,n is barycentric.

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