Non-integral formulas for brick semiorders beyond ladders

Derive explicit non-integral formulas for the asymptotic probabilities of brick semiorders that are not ladders in the uniform random unit-interval model; specifically, determine closed-form expressions for the leading coefficient C(P) in the relation L^n Pr(P generated on [0,L]) = C(P) L + O(1) as L → ∞, without relying on the multiple-integral representation F(P,L).

Background

The paper develops an integral method F(P,L) to approximate probabilities of generating a fixed semiorder P from n independent Uniform[0,L] points and proves that for bricks L^n Pr(P) equals a linear function in L up to a constant error term. For ladders (a special class of bricks), the leading coefficient is identified in terms of the Up/Down (Euler) numbers, yielding L^n Pr(P) ∼ n A_{n-1} L.

However, beyond ladders, the authors have not obtained closed-form expressions outside of integral representations for the leading asymptotic coefficient for bricks. The open problem asks for closed-form (non-integral) formulas—analogous to the ladder case—for general bricks that are not ladders.