Combinatorial theory of \(\text{T}\)-fractions and two points Padé approximants
In a well-known paper, \textit{Ph. Flajolet} [Discrete Math. 32, 125-161 (1980; Zbl 0445.05014)] has shown that the power series expansions on Jacobi and Stieltjes continued fractions could be interpreted combinatorially in terms of labelled paths in the plane (called respectively Motzkin and Dyck paths). In this paper, the authors propose a similar interpretation of Thorn's continued fractions which are of the form \[ {1\over 1- b_1 t-} {\lambda_1 t\over 1- b_2 t-}\cdots {\lambda_k t\over b_{k+ 1} t-}\cdots\;. \] This interpretation is in terms of labelled Dyck paths, where the range of the labels depends on whether they correspond to a local maximum of the Dyck path or not. It is shown that the Padé approximation property of Thorn's continued fractions (they approximate simultaneously a power series in \(t\) and a power series in \(t^{- 1}\)) can be derived from the combinatorics of Dyck paths. They exhibit a bijection between certain labelled Dyck paths (called Hermite histories) and permutations, from which they infer some continued fraction expansion for the generating series of a trivariate statistics on permutations. Similar bijections had been given by \textit{J. Françon} and the second author [Discrete Math. 28, 21-35 (1979; Zbl 0409.05003)], and by \textit{D. Foata} and \textit{D. Zeilberger} [Stud. Appl. Math. 83, No. 1, 31-59 (1990; Zbl 0738.05001)], related to Jacobi and Stieltjes continued fractions respectively. Finally, some applications to polyomino evaluation are given.
scientific article; zbMATH DE number 897610