Michael Somos conjectured a relation between Hankel determinants whose entries $\frac 1{2n+1}\binom{3n}n$ count ternary trees and the number of certain plane partitions and alternating sign matrices. Tamm evaluated these determinants by showing that the generating function for these entries has a continued fraction that is a special case of Gauss's continued fraction for a quotient of hypergeometric series. We give a systematic application of the continued fraction method to a number of similar Hankel determinants. We also describe a simple method for transforming determinants using the generating function for their entries. In this way we transform Somos's Hankel determinants to known determinants, and we obtain, up to a power of 3, a Hankel determinant for the number of alternating sign matrices. We obtain a combinatorial proof, in terms of nonintersecting paths, of determinant identities involving the number of ternary trees and more general determinant identities involving the number of $r$-ary trees.
We derive continued fractions for partition generating functions, utilizing both Euler's techniques and Ramanujan's techniques. Although our results are for integer partitions there is scope to extend this work to vector partitions, including for binary and n-ary partitions.
In this note we consider ternary trees naturally embedded in the plane in a deterministic way such that the root has position zero, or in other words label zero, and the children of a node with position $j$ have positions $j-1$, $j$, and $j+1$, for all $j\in\Z$. We derive the generating function of ternary trees where all nodes have labels which are less or equal than $j$, with $j\in\N$, and the generating function of ternary trees counted with respect to nodes with label $j$, with $j\in\Z$. Moreover, we discuss generalizations of the counting problem to several labels at the same time. Furthe…
This paper considers the enumeration of ternary trees (i.e. rooted ordered trees in which each vertex has 0 or 3 children) avoiding a contiguous ternary tree pattern. We begin by finding recurrence relations for several simple tree patterns; then, for more complex trees, we compute generating functions by extending a known algorithm for pattern-avoiding binary trees. Next, we present an alternate one-dimensional notation for trees which we use to find bijections that explain why certain pairs of tree patterns yield the same avoidance generating function. Finally, we compare our bijections to k…
We introduce the hydra continued fractions, as a generalization of the Rogers-Ramanujan continued fractions, and give a combinatorial interpretation in terms of shift-plethystic trees. We then show it is possible to express them as a quotient of m-distinct partition generating functions, and in its dual form as a quotient of the generating functions of compositions with contiguous rises upper bounded by m-1. We obtain new generating functions for compositions according to their local minima, for partitions with a prescribed set of rises, and for compositions with prescribed sets of contiguous …
Using the slow triangle map (a type of multi-dimensional continued fraction algorithm), we exhibit a method for generating any number of new identities for subsets of integer partitions.
Several authors have examined connections among restricted permutations, continued fractions, and Chebyshev polynomials of the second kind. In this paper we prove analogues of these results for involutions which avoid 3412. Our results include a recursive procedure for computing the generating function for involutions which avoid 3412 and any set of additional patterns. We use our results to give enumerations and generating functions for involutions which avoid 3412 and various sets of additional patterns. In many cases we express these generating functions in terms of Chebyshev polynomials of…
We present family of automatic sequences that define algebraic continued fractions in characteristic 2. This family is constructed from ultimately period words and contains the period-doubling sequence.
Two subclasses of Motzkin paths, S-Motzkin and T-Motzkin paths, are introduced. We provide bijections between S-Motzkin paths and ternary trees, S-Motzkin paths and non-crossing trees, and T-Motzkin paths and ordered pairs of ternary trees. Symbolic equations for both paths, and thus generating functions for the paths, are provided. Using these, various parameters involving the two paths are analyzed.
We find a generating function expressed as a continued fraction that enumerates ordered trees by the number of vertices at different levels. Several Catalan problems are mapped to an ordered-tree problem and their generating functions also expressed as a continued fraction. Among these problems is the enumeration of (132)-pattern avoiding permutations that have a given number of increasing patterns of length k. This extends and illuminates a result of Robertson, Wilf and Zeilberger for the case k=3.
Several authors have examined connections between permutations which avoid 132, continued fractions, and Chebyshev polynomials of the second kind. In this paper we prove analogues of some of these results for permutations which avoid 1243 and 2143. Using tools developed to prove these analogues, we give enumerations and generating functions for permutations which avoid 1243, 2143, and certain additional patterns. We also give generating functions for permutations which avoid 1243 and 2143 and contain certain additional patterns exactly once. In all cases we express these generating functions i…
We show that certain Riordan arrays have generating functions that can be expressed as continued fractions of Jacobi and Thron type. We investigate the inverses of such arrays, which in certain circumstances can also have generating functions representable as continued fractions. Links to orthogonal polynomial moment sequences, and to Laurent biorthogonal polynomials are developed. We show that certain Riordan group involutions can be defined by continued fractions. We also show how simple transformations of the Jacobi continued fractions can lead to exponential Riordan arrays. Finally, by way…
The article studies a class of generalized factorial functions and symbolic product sequences through Jacobi type continued fractions (J-fractions) that formally enumerate the divergent ordinary generating functions of these sequences. The more general definitions of these J-fractions extend the known expansions of the continued fractions originally proved by Flajolet that generate the rising factorial function, or Pochhammer symbol, $(x)_n$, at any fixed non-zero indeterminate $x \in \mathbb{C}$. The rational convergents of these generalized J-fractions provide formal power series approximati…
We study generating functions for the number of involutions, even involutions, and odd involutions in $S_n$ subject to two restrictions. One restriction is that the involution avoid 3412 or contain 3412 exactly once. The other restriction is that the involution avoid another pattern $τ$ or contain $τ$ exactly once. In many cases we express these generating functions in terms of Chebyshev polynomials of the second kind.
We investigate combinatorial properties of a kind of insets we defined in an earlier paper, interpreting them now in terms of restricted ternary words. This allows us to give new combinatorial interpretations of a number of known integer sequences, namely the coefficients of Chebyshev polynomials of both kinds, Fibonacci numbers, Delannoy numbers, asymmetric Delannoy numbers, Sulanke numbers, coordinating sequences for some cubic lattices, crystal ball sequences for some cubic lattices, and others. We also obtain several new properties of said insets. In particular, we derive three generating …
We study generating functions for the number of permutations in $S_n$ subject to set of restrictions. One of the restrictions belongs to $S_3$, while the others to $S_k$. It turns out that in a large variety of cases the answer can be expressed via continued fractions, and Chebyshev polynomials of the second kind.
We prove two new forms of Jacobi-type J-fraction expansions generating the binomial coefficients, $\binom{x+n}{n}$ and $\binom{x}{n}$, over all $n \geq 0$. Within the article we establish new forms of integer congruences for these binomial coefficient variations modulo any (prime or composite) $h \geq 2$ and compare our results with existing known congruences for the binomial coefficients modulo primes $p$ and prime powers $p^k$. We also prove new exact formulas for these binomial coefficient cases from the expansions of the $h^{th}$ convergent functions to the infinite J-fraction series gener…
We continue the investigations of lattice walks in the three dimensional lattice restricted to the positive octant. We separate models which clearly have a D-finite generating function from models for which there is no reason to expect that their generating function is D-finite, and we isolate a small set of models whose nature remains unclear and requires further investigation. For these, we give some experimental results about their asymptotic behaviour, based on the inspection of a large number of initial terms. At least for some of them, the guessed asymptotic form seems to tip the balance…
Sulanke and Xin developed a continued fraction method that applies to evaluate Hankel determinants corresponding to quadratic generating functions. We use their method to give short proofs of Cigler's Hankel determinant conjectures, which were proved recently by Chang-Hu-Zhang using direct determinant computation. We find that shifted periodic continued fractions arise in our computation. We also discover and prove some new nice Hankel determinants relating to lattice paths with step set $\{(1,1),(q,0), (\ell-1,-1)\}$ for integer parameters $m,q,\ell$. Again shifted periodic continued fraction…
In a recent paper G. Bhatnagar has given simple proofs of some of Ramanujan's continued fractions. In this note we show that some variants of these continued fractions are generating functions of q-Schroeder-like numbers.
Using ordinary and exponential generating functions, we explore the reversion of power series defined by $2$nd order recurrences. We express the reversions in terms of Jacobi and Thron continued fractions. We find relations with Eulerian expressions using a transformation of continued fractions.
We use a continued fraction approach to compare two statistical ensembles of quadrangulations with a boundary, both controlled by two parameters. In the first ensemble, the quadrangulations are bicolored and the parameters control their numbers of vertices of both colors. In the second ensemble, the parameters control instead the number of vertices which are local maxima for the distance to a given vertex, and the number of those which are not. Both ensembles may be described either by their (bivariate) generating functions at fixed boundary length or, after some standard slice decomposition, …
Let f_n^r(k) be the number of 132-avoiding permutations on n letters that contain exactly r occurrences of 12... k, and let F_r(x;k) and F(x,y;k) be the generating functions defined by $F_r(x;k)=\sum_{n\gs0} f_n^r(k)x^n$ and $F(x,y;k)=\sum_{r\gs0}F_r(x;k)y^r$. We find an explcit expression for F(x,y;k) in the form of a continued fraction. This allows us to express F_r(x;k) for $1\ls r\ls k$ via Chebyshev polynomials of the second kind.
We define a three parameter family of Bell pseudo-involutions in the Riordan group. The defining sequences have generating functions that are expressible as continued fractions. We indicate that the Hankel transforms of the defining sequences, and of the $A$ sequences of the corresponding Riordan arrays, can be associated with Somos $4$ sequence. We give examples where these sequences can be associated with elliptic curves, and we exhibit instances where elliptic curves can give rise to associated Riordan pseudo-involutions. In the case of a particular one parameter family of elliptic curves, …
The lattice path model suggested by E. Deutsch is derived from ordinary Dyck paths, but with additional down-steps of size -3,-5,-7,... . For such paths, we find the generating functions of them, according to length, ending at level $i$, both, when considering them from left to right and from right to left. The generating functions are intrinsically cubic, and thus (for $i=0$) in bijection to various objects, like even trees, ternary trees, etc.
We find a Thron-type continued fraction (T-fraction) for the ordinary generating function of the Ward polynomials, as well as for some generalizations employing a large (indeed infinite) family of independent indeterminates. Our proof is based on a bijection between super-augmented perfect matchings and labeled Schröder paths, which generalizes Flajolet's bijection between perfect matchings and labeled Dyck paths.
A permutation can be locally classified according to the four local types: peaks, valleys, double rises and double falls. The corresponding classification of binary increasing trees uses four different types of nodes. Flajolet demonstrated the continued fraction representation of the generating function of local types, using a classical bijection between permutations, binary increasing trees, and suitably defined path diagrams induced by Motzkin paths. The aim of this article is to extend the notion of local types from permutations to $k$-Stirling permutations (also known as $k$-multipermutati…
A plateau in a Motzkin path is a sequence of three steps: an up step, a horizontal step, then a down step. We find three different forms for the bivariate generating function for plateaus in Motzkin paths, then generalize to longer plateaus. We conclude by describing a further generalization: a continued fraction form from which one can easily derive new multivariate generating functions for various kinds of path statistics. Several examples of generating functions are given using this technique.
It is well known that the $(-1)$-evaluation of the enumerator polynomials of permutations (resp. derangements) by the number of excedances gives rise to tangent numbers (resp. secant numbers). Recently, two distinct $q$-analogues of the latter result have been discovered by Foata and Han, and Josuat-Vergès, respectively. In this paper, we will prove some general continued fractions expansions formulae, which permits us to give a unified treatment of Josuat-Vergès' two formulae and also to derive a new $q$-analogue of the aforementioned formulae. Our approach is based on a $(p,q)$-analogue of t…
In [BS] Babson and Steingrimsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Let $f_{τ;r}(n)$ be the number of $1\mn3\mn2$-avoiding permutations on $n$ letters that contain exactly $r$ occurrences of $τ$, where $τ$ a generalized pattern on $k$ letters. Let $F_{τ;r}(x)$ and $F_τ(x,y)$ be the generating functions defined by $F_{τ;r}(x)=\sum_{n\geq0} f_{τ;r}(n)x^n$ and $F_τ(x,y)=\sum_{r\geq0}F_{τ;r}(x)y^r$. We find an explicit expression for $F_τ(x,y)$ in the form of a continued fraction for whe…
The main objects of the investigation presented in this paper are branched-continued-fraction representations of ratios of contiguous hypergeometric series and type II multiple orthogonal polynomials on the step-line with respect to linear functionals or measures whose moments are ratios of products of Pochhammer symbols. This is an interesting case study of the recently found connection between multiple orthogonal polynomials and branched continued fractions that gives a clear example of how this connection leads to considerable advances on both topics. We obtain new results about generating …
The goal of this paper is twofold. First, we review the recently developed geometric approach to the combinatorics of the median Genocchi numbers. The Genocchi numbers appear in this context as Euler characteristics of the degenerate flag varieties. Second, we prove that the generating function of the Poincar\' e polynomials of the degenerate flag varieties can be written as a simple continued fraction. As an application we prove that the Poincar\' e polynomials coincide with the $q$-version of the normalized median Genocchi numbers introduced by Han and Zeng.
We find, in the form of a continued fraction, the generating function for the number of (132)-avoiding permutations that have a given number of (123) patterns, and show how to extend this to permutations that have exactly one (132) pattern. We find some properties of the continued fraction, which is similar to, though more general than, those that were studied by Ramanujan.
We study the triangular array defined by the Graham--Knuth--Patashnik recurrence $T(n,k) = (αn + βk + γ)\, T(n-1,k)+(α' n + β' k + γ') \, T(n-1,k-1)$ with initial condition $T(0,k) = δ_{k0}$ and parameters $\mathbfμ = (α,β,γ, α',β',γ')$. We show that the family of arrays $T(\mathbfμ)$ is invariant under a 48-element discrete group isomorphic to $S_3 \times D_4$. Our main result is to determine all parameter sets $\mathbfμ \in \mathbb{C}^6$ for which the ordinary generating function $f(x,t) = \sum_{n,k=0}^\infty T(n,k) \, x^k t^n$ is given by a Stieltjes-type continued fraction in $t$ with coef…
We introduce the generic Lah polynomials $L_{n,k}(ϕ)$, which enumerate unordered forests of increasing ordered trees with a weight $ϕ_i$ for each vertex with $i$ children. We show that, if the weight sequence $ϕ$ is Toeplitz-totally positive, then the triangular array of generic Lah polynomials is totally positive and the sequence of row-generating polynomials $L_n(ϕ,y)$ is coefficientwise Hankel-totally positive. Upon specialization we obtain results for the Lah symmetric functions and multivariate Lah polynomials of positive and negative type. The multivariate Lah polynomials of positive typ…
We define and characterize the $f$-matrices associated to Pascal-like matrices that are defined by ordinary and exponential Riordan arrays. These generalize the face matrices of simplices and hypercubes. Their generating functions can be expressed simply in terms of continued fractions, which are shown to be transformations of the generating functions of the corresponding $γ$- and $h$-matrices.
In the theory of continued fractions, Zaremba's conjecture states that there is a positive integer $M$ such that each integer is the denominator of a convergent of an ordinary continued fraction with partial quotients bounded by $M$. In this paper, to each such $M$ we associate a regular sequence---in the sense of Allouche and Shallit---and establish various properties and results concerning the generating function of the regular sequence. In particular, we determine the minimal algebraic relation concerning the generating function and its Mahler iterates.
Many combinatorial numbers can be placed in the following generalized triangular array $[T_{n,k}]_{n,k\ge 0}$ satisfying the recurrence relation: \begin{equation*} T_{n,k}=λ(a_0n+a_1k+a_2)T_{n-1,k}+(b_0n+b_1k+b_2)T_{n-1,k-1}+\frac{d(da_1-b_1)}λ(n-k+1)T_{n-1,k-2} \end{equation*} with $T_{0,0}=1$ and $T_{n,k}=0$ unless $0\le k\le n$ for suitable $a_0,a_1,a_2,b_0,b_1,b_2,d$ and $λ$. For $n\geq0$, denote by $T_n(q)$ the generating function of the $n$-th row. In this paper, we develop various criteria for $\textbf{x}$-Stieltjes moment property and $3$-$\textbf{x}$-log-convexity of $T_n(q)$ based on…
In this review article, we present a unified approach to solving discrete, integrable, possibly non-commutative, dynamical systems, including the $Q$- and $T$-systems based on $A_r$. The initial data of the systems are seen as cluster variables in a suitable cluster algebra, and may evolve by local mutations. We show that the solutions are always expressed as Laurent polynomials of the initial data with non-negative integer coefficients. This is done by reformulating the mutations of initial data as local rearrangements of continued fractions generating some particular solutions, that preserve…
This study presents miscellaneous properties of pseudo-factorials, which are numbers whose recurrence relation is a twisted form of that of usual factorials. These numbers are associated with special elliptic functions, most notably, a Dixonian and a Weierstrass function, which parametrize the Fermat cubic curve and are relative to a hexagonal lattice. A continued fraction expansion of the ordinary generating function of pseudo-factorials, first discovered empirically, is established here. This article also provides a characterization of the associated orthogonal polynomials, which appear to f…
Dumont and Foata introduced in 1976 a three-variable symmetric refinement of Genocchi numbers, which satisfies a simple recurrence relation. A six-variable generalization with many similar properties was later considered by Dumont. They generalize a lot of known integer sequences, and their ordinary generating function can be expanded as a Jacobi continued fraction. We give here a new combinatorial interpretation of the six-variable polynomials in terms of the alternative tableaux introduced by Viennot. A powerful tool to enumerate alternative tableaux is the so-called "matrix Ansatz", and usi…
We note that a tridiagonal matrix representation of the algebra of the partially asymmetric exclusion process (PASEP) lends itself to interpretation as the transfer matrix for weighted Motzkin lattice paths. A continued fraction ("J-Fraction") representation of the lattice path generating function is particularly well suited to discussing the PASEP, for which the paths have height dependent weights. We show that this not only allows a succinct derivation of the normalisation and correlation lengths of the PASEP, but also reveals how finite-dimensional representations of the PASEP algebra, vali…
In a 1977 paper, Diaconis and Graham studied what Knuth calls the total displacement of a permutation $w$, which is the sum of the distances $|w(i)-i|$. In recent work of the first author and Tenner, this statistic appears as twice the type $A_{n-1}$ version of a statistic for Coxeter groups called the depth of $w$. There are various enumerative results for this statistic in the work of Diaconis and Graham, codified as exercises in Knuth's textbook, and some other results in the work of Petersen and Tenner. However, no formula for the generating function of this statistic appears in the litera…
We study the generating function of descent numbers for the permutations with descent pairs of prescribed parities, the distribution of which turns out to be a refinement of median Genocchi numbers. We prove the $γ$-positivity for the polynomial and derive the generating function for the $γ$-vectors, expressed in the form of continued fraction. We also come up with an artificial statistic that gives a $q$-analogue of the $γ$-positivity for the permutations with descents only allowed from an odd value to an odd value.
In this paper we prove the strong $q$-log-convexity of the Eulerian polynomials of Coxeter groups using their exponential generating functions. Our proof is based on the theory of exponential Riordan arraya and a criterion for determining the strong $q$-log-convexity of polynomials sequences, whose generating functions can be given by the continued fraction. As consequences, we get the strong $q$-log-convexity the Eulerian polynomials of type $A_n,B_n$, their $q$-analogous and the generalized Eulerian polynomials associated to the arithmetic progression $\{a,a+d,a+2d,a+3d,\ldots\}$ in a unifie…
We define an infinite sequence of generalizations, parametrized by an integer $m \ge 1$, of the Stieltjes--Rogers and Thron--Rogers polynomials; they arise as the power-series expansions of some branched continued fractions, and as the generating polynomials for $m$-Dyck and $m$-Schröder paths with height-dependent weights. We prove that all of these sequences of polynomials are coefficientwise Hankel-totally positive, jointly in all the (infinitely many) indeterminates. We then apply this theory to prove the coefficientwise Hankel-total positivity for combinatorially interesting sequences of …
We study joint distributions of cycles and patterns in permutations written in standard cycle form. We explore both classical and generalised patterns of length 2 and 3. Many extensions of classical theory are achieved; bivariate generating functions for inversions, ascents, descents, 123s, valleys, 1'-2-1s; closed forms forms for avoidance of peaks, 2-3-1s, 1-2-3s, 2'-1-2s and 1'-2-1s; bijective proofs of Wilf-equivalences. We also derive some results about standard pattern occurrence, such as continued fractions for the generating functions for occurrences of valleys and the pattern 123. The…
This work describes a classification of the $N$-soliton solutions of the Kadomtsev-Petviashvili II equation in terms of chord diagrams of N chords joining pairs of 2N points. The different classes of N-solitons are enumerated by the distribution of crossings of the chords. The generating function of the chord diagrams is expressed as a continued fraction, special cases of which are moment generating functions for certain kinds of $q$-orthogonal polynomials.
We derive the continued fraction form of the generating function of some new $q$-analogs of the Eulerian numbers $\hat{E}_{k,n}(q)$ introduced by Lauren Williams building on work of Alexander Postnikov. They are related to the number of alignments and weak exceedances of permutations. We show how these numbers are related to crossing and generalized patterns of permutations We generalize to the case of decorated permutations. Finally we show how these numbers appear naturally in the stationary distribution of the ASEP model.
We introduce some new classes of words and permutations characterized by the second difference condition $π(i-1) + π(i+1) - 2π(i) \leq k$, which we call the $k$-convexity condition. We demonstrate that for any sized alphabet and convexity parameter $k$, we may find a generating function which counts $k$-convex words of length $n$. We also determine a formula for the number of 0-convex words on any fixed-size alphabet for sufficiently large $n$ by exhibiting a connection to integer partitions. For permutations, we give an explicit solution in the case $k = 0$ and show that the number of 1-conve…
