Exponential generating functions for ladders with twins in higher positions

Determine an explicit exponential generating function for the coefficient sequence governing ladder semiorders with a twin in position i ≥ 3; concretely, find a generating function G_i(x) such that, for the ladder on n elements with a twin in the i-th position (defined so that removing one twin yields the ladder on n−1 elements and the remaining twin occupies position i in every endpoint linear extension), the asymptotic relation L^n Pr(P) ∼ n B_{n-1} L holds with {B_n} having exponential generating function G_i(x).

Background

The authors paper ladders with a single twin positioned at a specified location in every endpoint linear extension. Based on computed probabilities, they propose explicit exponential generating functions for the first two positions: (x−1)(sec x + tan x) for a twin in position 1 and (2−x)(sec x + tan x) for a twin in position 2, and they report agreement up to n = 20.

For positions i ≥ 3, no exponential generating function is currently known; natural variations of the earlier formulas do not match the computed data. The open problem is to identify the correct exponential generating functions in these cases.