Efficient computation of double-deficiency generating functions for 123-avoiding permutations

Find an efficient method for computing the double-deficiency generating functions f_{123}(n,x)=\sum_{\pi\in Av_n(123)}x^{\operatorname{DD}(\pi)} for permutations avoiding the pattern 123.

Background

The paper develops generating-function recurrences for the distribution of double deficiencies in permutations avoiding individual patterns of length 3. Structural decompositions handle the classes Av(132), Av(213), and Av(231), while a lattice-path method yields an algebraic generating function for Av(321); the statistic vanishes identically in Av(312). The class Av(123), however, does not admit an analogous decomposition in the paper, and a potential path-based approach does not align naturally with the double-deficiency condition.

The authors explicitly state that they have not obtained a closed recurrence for the generating function of double deficiencies in Av(123). The problem therefore asks for an efficient computational method, such as a recurrence or another explicit generating-function framework, for this remaining single-pattern case.