The Generating Function of Ternary Trees and Continued Fractions

We introduce a new family 𝒜 n,k of Schur positive symmetric functions, which are defined as sums over totally symmetric plane partitions. In the first part, we show that, for k=1, this family is equal to a multivariate generating function involving n+3 variables of objects that extend alternating sign matrices (ASMs), which have recently been introduced by the authors. This establishes a new connection between ASMs and a class of plane partitions, thereby complementing the fact that ASMs are equinumerous with totally symmetric self-complementary plane partitions as well as with descending plane partitions. The proof is based on a new antisymmetrizer-to-determinant formula for which we also provide a combinatorial proof, and, although this proof is complicated, it is an important step forward as it is very hard to find combinatorial proofs in this field. In the second part, we relate three specialisations of 𝒜 n,k to weighted enumerations of certain well-known classes of column strict shifted plane partitions that generalise descending plane partitions.

Hankel continued fraction and its applications

Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We put ASMs into a larger context by studying the order ideals of subposets of a certain poset, proving that they are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self―complementary plane partitions (TSSCPPs), Catalan objects, tournaments, semistandard Young tableaux, and totally symmetric plane partitions. We use this perspective to prove an expansion of the tournament generating function as a sum over TSSCPPs which is analogous to a known formula involving ASMs. Les matrices à signe alternant (ASMs) sont des matrices carrées dont les coefficients sont 0,1 ou -1, telles que dans chaque ligne et chaque colonne la somme des entrées vaut 1 et les entrées non nulles ont des signes qui alternent. Nous incluons les ASMs dans un cadre plus vaste, en étudiant les idéaux des sous-posets d'un certain poset, dont nous prouvons qu'ils sont en bijection avec de nombreux objets combinatoires intéressants, tels que les ASMs, les partitions planes totalement symétriques autocomplémentaires (TSSCPPs), des objets comptés par les nombres de Catalan, les tournois, les tableaux semistandards, ou les partitions planes totalement symétriques. Nous utilisons ce point de vue pour démontrer un développement de la série génératrice des tournois en une somme portant sur les TSSCPPs, analogue à une formule déjà connue faisant appara\^ıtre les ASMs.

The six-vertex model with domain wall boundary conditions, or squareice, is considered for particular values of its parameters, corresponding to1-,2-,and 3-enumerations of alternating sign matrices (ASMs). Using Hankel determinant representationsfor the partition function and the boundary correlator of homogeneous square ice, it is shownhow the ordinary and refined enumerations can be derived in a very simple and straightforwardway. The derivation is based on the standard relationship between Hankel determinantsand orthogonal polynomials. For the particular sets of parameters corresponding to1-,2-,and 3-enumerations of ASMs, the Hankel determinant can be naturally related to ContinuousHahn, Meixner–Pollaczek, and Continuous Dual Hahn polynomials, respectively. Thisobservation allows for a unified and simplified treatment of ASM enumerations.In particular, along the lines of the proposed approach, we provide a completesolution to the long-standing problem of the refined 3-enumeration of ASMs.

We consider straight-line drawings of trees on a hexagonal grid. The hexagonal grid is an extension of the common grid with inner nodes of degree six. We restrict the number of directions used for the edges from each node to its children from one to five, and to five patterns: straight, Y, ψ, X, and full. The ψ–drawings generalize hv- or strictly upward drawings to ternary trees. We show that complete ternary trees have a ψ–drawing on a square of size $\O(n^{1.262})$ and general ternary trees can be drawn within $\O(n^{1.631})$ area. Both bounds are optimal. Sub–quadratic bounds are also obtained for X–pattern drawings of complete tetra trees, and for full–pattern drawings of complete penta trees, which are 4–ary and 5–ary trees. These results parallel and complement the ones of Frati [8] for straight–line orthogonal drawings of ternary trees. Moreover, we provide an algorithm for compacted straight–line drawings of penta trees on the hexagonal grid, such that the direction of the edges from a node to its children is given by our patterns and these edges have the same length. However, drawing trees on a hexagonal grid within a prescribed area or with unit length edges is $\mathcal{NP}$–hard.

The paper deals with the problem of representation of Horn’s hypergeometric functions via continued fractions and branched continued fractions. We construct the formal continued fraction expansions for three ratios of Horn’s hypergeometric functions H7. The method employed is a two-dimensional generalization of the classical method of constructing a Gaussian continued fraction. It is proved that the continued fraction, which is an expansion of each ratio, uniformly converges to a holomorphic function of two variables on every compact subset of some domain of C2, and that this function is an analytic continuation of such a ratio in this domain. To illustrate this, we provide some numerical experiments at the end.

In this paper, we present two different versions of Vincent’s theorem of 1836 and discuss various real root isolation methods derived from them: one using continued fractions and two using bisections, the former being the fastest real root isolation method. Regarding the continued fractions method, we first show how, using a recently developed quadratic complexity bound on the values of the positive roots of polynomials, its performance has been improved by an average of 40% over its initial implementation, and then we indicate directions for future research. Bibliography: 45 titles.

We demonstrate a natural bijection between a subclass of alternating sign matrices (ASMs) defined by a condition on the corresponding monotone triangle which we call the gapless condition and a subclass of totally symmetric self-complementary plane partitions defined by a similar condition on the corresponding fundamental domains or Magog triangles. We prove that, when restricted to permutations, this class of ASMs reduces to 312-avoiding permutations. This leads us to generalize pattern avoidance on permutations to a family of words associated to ASMs, which we call Gog words. We translate the gapless condition on monotone triangles into a pattern avoidance-like condition on Gog words associated. We estimate the number of gapless monotone triangles using a bijection with $p$-branchings.

Short proof of the ASM theorem avoiding the six-vertex model

The number of shifted plane partitions contained in the shifted shape $[p + q - 1,p + q - 3, \ldots ,p - q + 1]$ with part size bounded by $m$ is shown to be equal to the number of ordinary plane partitions contained in the shape $(p,p, \ldots ,p)$ $(q{\text { rows}})$ with part size bounded by $m$. The proof uses known combinatorial descriptions of finite-dimensional representations of semisimple Lie algebras. A separate simpler argument shows that the number of chains of cardinality $k$ in the poset underlying the shifted plane partitions is equal to the number of chains of cardinality $k$ in the poset underlying the ordinary plane partitions. The first result can also be formulated as an equality of chain counts for a pair of posets. The pair of posets is obtained by taking order ideals in the other pair of posets.

Assuming that a plane partition of the positive integer n is chosen uniformly at random from the set of all such partitions, we propose a general asymptotic scheme for the computation of expectations of various plane partition statistics as n becomes large. The generating functions that arise in this study are of the form Q(x)F(x), where $$Q(x)=\prod _{j=1}^\infty (1-x^j)^{-j}$$ is the generating function for the number of plane partitions. We show how asymptotics of such expectations can be obtained directly from the asymptotic expansion of the function F(x) around $$x=1$$ . The representation of a plane partition as a solid diagram of volume n allows interpretations of these statistics in terms of its dimensions and shape. As an application of our main result, we obtain the asymptotic behavior of the expected values of the largest part, the number of columns, the number of rows (that is, the three dimensions of the solid diagram) and the trace (the number of cubes in the wall on the main diagonal of the solid diagram). Our results are similar to those of Grabner et al. (Comb Probab Comput 23:1057–1086, 2014) related to linear integer partition statistics. We base our study on the Hayman’s method for admissible power series.

It has recently been observed empirically that the number of FPL configurations with 3 sets of $a$, $b$ and $c$ nested arches equals the number of plane partitions in a box of size $a\times b \times c$. In this note, this result is proved by constructing explicitly the bijection between these FPL and plane partitions.

Wave propagation in a two-dimensional unbounded domain with rigid bedrock was studied. The study was based on the scaled-boundary finite-element method (SBFEM) combing the continued fraction method. The SBFEM is a novel semianalytical technique, combining the advantages of the finite-element and the boundary-element methods. Meanwhile, SBFEM has unique properties of its own. In this paper, the modified SBFEM was introduced. The original scaling center was replaced by a scaling line, which is more suitable for analyzing the multilayered soil model. As a key point, the modified SBFEM and the dual system, which is used to solve a proposed time domain problem, were combined to derive the governing equations. The derivation process was based on the framework of the Hamiltonian system by introducing dual variables. By using the continued fraction solution and introducing auxiliary variables, the equation of motion was built. More significantly, the efficient, precise time-integration method was first us...

A detailed investigation has been carried out of the relaxation of fluctuations in the steady state of the Stratonovich model, also known as the random growing rate model. The autocorrelation time of this system, driven by parametric white noise in the linear term, has been measured for an electronic circuit model, computed by digital simulation and calculated by use of a continued fraction expansion method. The results are all consistent with each other, and with those obtained from the matrix continued fraction method of Jung and Risken (1985), provided that explicit account is taken of the ways in which any macroscopic real physical system is bound to differ from the idealisation represented by the original form of the Stratonovich model. In particular, it is necessary to recognise the existence of weak additive noise and a small additive constant. The physical origins of some earlier, seemingly discrepant, calculations and experimental data are discussed.

The direct continued fraction method of model reduction for single-input-output and multi-input—output cases sometimes produces unstable reduced models. A generalization is made by using the squared magnitude continued fraction and factorization technique to obtain reduced models which are always stable.