Narrow Results

A heap ordered tree of size n is a planted plane tree together with a bijection from the nodes to the set {1,…,n} which is monotonically increasing when going from the root to the leaves. In a recent paper by Chen and Ni, the expectation and the variance of the depth of a random node in a random heap ordered tree of size n was considered. Here, we give additional results concerning level polynomials. From a historical point of view, we trace these trees back to an earlier paper by Prodinger and Urbanek and find formulae that are proved in the paper by Chen and Ni by ad hoc computations already in a classic book by Greene and Knuth. Also, we mention that a paper by Bergeron, Flajolet and Salvy develops a more general theory of increasing trees, based on simply generated families of trees. Furthermore we consider the path length which is a natural concept when dealing with the depth. Expectation and variance are obtained, both explicitly and asymptotically.

For a directed graph G, we generalize the Catalan numbers by using the canonical generating partial isometries of the Cuntz–Krieger algebra for the transition matrix A^G of the directed edges of G. The generalized Catalan numbers enumerate the number of Dyck paths for the graph G. Its generating functions will be studied.

We consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology: for a compact surface $S$, with a finite set of points $F$ fixed on its boundary, how many configurations of disjoint arcs are there on $S$ whose boundary is $F$? We find that this enumerative problem, counting curves on surfaces, has a rich structure. We show that such curve counts obey an effective recursion, in the general spirit of topological recursion, and exhibit quasi-polynomial behavior. This “elementary curve-counting” is in fact related to a more advanced notion of “curve-counting” from algebraic geometry or symplectic geometry. The asymptotics of this enumerative problem are closely related to the asymptotics of volumes of moduli spaces of curves, and the quasi-polynomials governing the enumerative problem encode intersection numbers on moduli spaces. Among several other results, we show that generating functions and differential forms for these curve counts exhibit structure that is reminiscent of the mathematical physics of free energies, partition functions and quantum curves.

We consider a model of margin trading based on the hyperfinite timeline. Using only elementary nonstandard analysis we are able to derive explicit formulas for the expected margin call time and loss. Further margin trading strategy is studied and an application to pricing barrier option is given. We prove a generalization of the Catalan numbers which forms the combinatoric basis of our results and should be of independent interest.

We define a deformation of free creations (and annihilations), given by operators on the full Fock space, acting nontrivially only between the vacuum subspace ℂΩ and the twofold tensor product . Then we study the distribution of the deformed free gaussian operators, with the deformation containing also a real parameter d. The recurrence formula for moments is shown, and the Cauchy transform of the distribution measure is computed. This yields the description of the measure: absolutely continuous part and the atomic part. The existence of atoms depends on the parameter d. The special case d =1 is studied with all details, with the formula for moments is given as values of the hypergeometric series. Finally we show the formula for computing the mixed moments of the deformed operators.

We develop basic cluster theory from an elementary point of view using a variation of binary trees which we call mixed cobinary trees (MCTs). We show that the number of isomorphism classes of such trees is given by the Catalan number $C_n$ where $n$ is the number of internal nodes. We also consider the corresponding quiver $Q_{𝜖}$ of type $A_{n-1}$. As a special case of more general known results about the relation between $c$-vectors, representations of quivers and their semi-invariants, we explain the bijection between MCTs and the vertices of the generalized associahedron corresponding to the quiver $Q_{𝜖}$. These results are extended to $m$-clusters in the next paper. We give one application: a new short proof of a conjecture of Reineke using MCTs.

In this paper we establish some new congruences involving central binomial coefficients as well as Catalan numbers. Let p be a prime and let a be any positive integer. We determine for d = 0, …, p^a and for δ = 0, 1. We also show that

For any prime p, we establish congruences modulo pⁿ⁺¹ for the difference of the pⁿ⁺¹th and pⁿth Motzkin numbers and determine the p-adic order of the difference. The results confirm recent conjectures on the order. The applied techniques involve the use of congruences for the differences of certain Catalan numbers and binomial coefficients, congruential identities for sums of Catalan numbers, central binomial and trinomial coefficients, infinite incongruent disjoint covering systems and the solution of congruential recurrences.

The theorems of Gauss and Jacobi that give modulo $p$ evaluations of certain central binomial coefficients have been extended, since the 1980s, to more classes of binomial coefficients and to congruences modulo $p^2$. In this paper, we further extend these results to congruences modulo $p^3$. In the process, we prove congruences to arbitrarily high powers of $p$ for certain quotients of Gauss factorials that resemble binomial coefficients and are related to Morita's $p$-adic gamma function. These congruences are of a simple form and involve Catalan numbers as coefficients.

We prove evaluations of Hankel determinants of linear combinations of moments of orthogonal polynomials (or, equivalently, of generating functions for Motzkin paths), thus generalizing known results for Catalan numbers.

We show that if ${\{U_n\}}_{n\ge 0}$ is a Lucas sequence, then the largest $n$ suc that $\left|U_n\right|=C_{m_1}{C}_{m_2}\cdots C_{m_k}$ with $1\le m_1\le m_2\le \cdots \le m_k$, where $C_m$ is the $m$th Catalan number satisfies $n<6500$. In case the roots of the Lucas sequence are real, we have $n\in \{1,2,3,4,6,8,12\}$. As a consequence, we show that if ${\{X_n\}}_{n\ge 1}$ is the sequence of the $X$ coordinates of a Pell equation $X^2-d{Y}^2=\pm 1$ with a nonsquare integer $d>1$, then $X_n=C_m$ implies $n=1$.

In the paper some trigonometric nature identities for finite sums of inverses of central binomial numbers and Catalan numbers are given.

It is well known that Catalan Numbers are used in Mathematical Biology as Enumerating RNA secondary structures for finite points. In this paper, we present some new properties of the Catalan Numbers.

We prove evaluations of Hankel determinants of linear combinations of moments of orthogonal polynomials (or, equivalently, of generating functions for Motzkin paths), thus generalizing known results for Catalan numbers.