arXiv subjects

Juan B. Gil

Publications and source records attributed to Juan B. Gil.

At least 37 records · Page 2Linked to original sources

arXiv2018

On factor-free Dyck words with half-integer slope

We study a class of rational Dyck paths with slope (2m+1)/2 corresponding to factor-free Dyck words, as introduced by P. Duchon. We show that, for the slopes considered in this paper, the language of factor-free Dyck words is generated by an auxiliary language that we examine from the algebraic and combinatorial points of view. We provide a lattice path description of this language, and give an explicit enumeration formula in terms of partial Bell polynomials. As a corollary, we obtain new formulas for the number of associated factor-free generalized Dyck words.

math.CO↗

arXiv2018

Compositions colored by simplicial polytopic numbers

For a given integer $d\ge 1$, we consider $\binom{n+d-1}{d}$-color compositions of a positive integer $ν$ for which each part of size $n$ admits $\binom{n+d-1}{d}$ colors. We give explicit formulas for the enumeration of such compositions, generalizing existing results for $n$-color compositions (case $d=1$) and $\binom{n+1}{2}$-color compositions (case $d=2$). In addition, we give bijections from the set of $\binom{n+d-1}{d}$-color compositions of $ν$ to the set of compositions of $(d+1)ν- 1$ having only parts of size $1$ and $d+1$, the set of compositions of $(d+1)ν$ having only parts of size congruent to $1$ modulo $d+1$, and the set of compositions of $(d+1)ν+ d$ having no parts of size less than $d+1$. Our results rely on basic properties of partial Bell polynomials and on a suitable adaptation of known bijections for $n$-color compositions.

math.CO↗

arXiv2017

Bounce statistics for rational lattice paths

Given two relatively prime positive integers $α$ and $β$, we consider simple lattice paths (with unit East and unit North steps) from $(0,0)$ to $(αk,βk)$, and enumerate them by their left and right bounces with respect to the line $y=\fracβα x$. We give the corresponding multivariate generating functions for all such paths as well as for subclasses of paths that start and end with a prescribed step. For illustration purposes, we discuss the case $β=1$ and express some of our functions in terms of the Fuss-Catalan generating function $c_α(x)$.

math.CO↗

arXiv2017

$(an+b)$-color compositions

For $a,b\in\mathbb{N}_0$, we consider $(an+b)$-color compositions of a positive integer $ν$ for which each part of size $n$ admits $an+b$ colors. We study these compositions from the enumerative point of view and give a formula for the number of $(an+b)$-color compositions of $ν$ with $k$ parts. Our formula is obtained in two different ways: 1) by means of algebraic properties of partial Bell polynomials, and 2) through a bijection to a certain family of weak compositions that we call domino compositions. We also discuss two cases when $b$ is negative and give corresponding combinatorial interpretations.

math.CO↗

arXiv2016

On rational Dyck paths and the enumeration of factor-free Dyck words

Motivated by independent results of Bizley and Duchon, we study rational Dyck paths and their subset of factor-free elements. On the one hand, we give a bijection between rational Dyck paths and regular Dyck paths with ascents colored by factor-free words. This bijection leads to a new statistic based on the reducibility level of the paths for which we provide a corresponding formula. On the other hand, we prove an inverse relation for certain sequences defined via partial Bell polynomials, and we use it to derive a formula for the enumeration of factor-free words. In addition, we give alternative formulas for various enumerative sequences that appear in the context of rational Dyck paths.

math.CO↗

arXiv2016

Enumeration of colored Dyck paths via partial Bell polynomials

We consider a class of lattice paths with certain restrictions on their ascents and down steps and use them as building blocks to construct various families of Dyck paths. We let every building block $P_j$ take on $c_j$ colors and count all of the resulting colored Dyck paths of a given semilength. Our approach is to prove a recurrence relation of convolution type, which yields a representation in terms of partial Bell polynomials that simplifies the handling of different colorings. This allows us to recover multiple known formulas for Dyck paths and related lattice paths in an unified manner.

math.CO↗

arXiv2016

On the enumeration of restricted words over a finite alphabet

We present a method for the enumeration of restricted words over a finite alphabet. Restrictions are described through the inclusion or exclusion of suitable building blocks used to construct the words by concatenation. Our approach, which relies on the invert transform and its representation in terms of partial Bell polynomials, allows us to generalize and address in a systematic manner previous results in the subject.

math.CO↗

arXiv2015

Linear recurrence sequences with indices in arithmetic progression and their sums

For an arbitrary homogeneous linear recurrence sequence of order d with constant coefficients, we derive recurrence relations for all subsequences with indices in arithmetic progression. The coefficients of these recurrences are given explicitly in terms of partial Bell polynomials that depend on at most d-1 terms of the generalized Lucas sequence associated with the given recurrence. We also provide an elegant formula for the partial sums of such sequences and illustrate all of our results with examples of various orders, including common generalizations of the Fibonacci numbers.

math.NT↗

arXiv2015

Colored partitions of a convex polygon by noncrossing diagonals

For any positive integers $a$ and $b$, we enumerate all colored partitions made by noncrossing diagonals of a convex polygon into polygons whose number of sides is congruent to $b$ modulo $a$. For the number of such partitions made by a fixed number of diagonals, we give both a recurrence relation and an explicit representation in terms of partial Bell polynomials. We use basic properties of these polynomials to efficiently incorporate restrictions on the type of polygons allowed in the partitions.

math.CO↗

arXiv2014

Convolutions of Tribonacci, Fuss-Catalan, and Motzkin sequences

We introduce a class of sequences, defined by means of partial Bell polynomials, that contains a basis for the space of linear recurrence sequences with constant coefficients as well as other well-known sequences like Catalan and Motzkin. For the family of `Bell sequences' considered in this paper, we give a general multifold convolution formula and illustrate our result with a few explicit examples.

math.NT↗

arXiv2014

Linear recurrence sequences and their convolutions via Bell polynomials

We recast homogeneous linear recurrence sequences with fixed coefficients in terms of partial Bell polynomials, and use their properties to obtain various combinatorial identities and multifold convolution formulas. Our approach relies on a basis of sequences that can be obtained as the INVERT transform of the coefficients of the given recurrence relation. For such a basis sequence $(y_n)$ with generating function $Y(t)$, and for any positive integer $r$, we give a formula for the convolved sequence generated by $Y(t)^r$ and prove that it satisfies an elegant recurrence relation.

math.CO↗

arXiv2013

On Hensel's roots and a factorization formula in Z[[x]]

Given an odd prime $p$, we provide formulas for the Hensel lifts of polynomial roots modulo $p$, and give an explicit factorization over the ring of formal power series with integer coefficients for certain reducible polynomials whose constant term is of the form $p^w$ with $w>1$. All of our formulas are given in terms of partial Bell polynomials and rely on the inversion formula of Lagrange.

math.NT↗

arXiv2012

Some convolution identities and an inverse relation involving partial Bell polynomials

We prove an inverse relation and a family of convolution formulas involving partial Bell polynomials. Known and some presumably new combinatorial identities of convolution type are discussed. Our approach relies on an interesting multinomial formula for the binomial coefficients. The inverse relation is deduced from a parametrization of suitable identities that facilitate dealing with compositions of Bell polynomials.

math.CO↗

arXiv2011

Factoring polynomials in the ring of formal power series over Z

We consider polynomials with integer coefficients and discuss their factorization properties in Z[[x]], the ring of formal power series over Z. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility as power series. Moreover, if a polynomial is reducible over Z[[x]], we provide an explicit factorization algorithm. For polynomials whose constant term is a prime power, our study leads to the discussion of p-adic integers.

math.AC↗

arXiv2010

On the closure of elliptic wedge operators

We prove a semi-Fredholm theorem for the minimal extension of elliptic operators on manifolds with wedge singularities and give, under suitable assumptions, a full asymptotic expansion of the trace of the resolvent.

math.AP↗

arXiv2009

Trace expansions for elliptic cone operators with stationary domains

We analyze the behavior of the trace of the resolvent of an elliptic cone differential operator as the spectral parameter tends to infinity. The resolvent splits into two components, one associated with the minimal extension of the operator, and another, of finite rank, depending on the particular choice of domain. We give a full asymptotic expansion of the first component and expand the component of finite rank in the case where the domain is stationary. The results make use, and develop further, our previous investigations on the analytic and geometric structure of the resolvent. The analysis of nonstationary domains, considerably more intricate, is pursued elsewhere.

math.SP↗

arXiv2007

A conic manifold perspective of elliptic operators on graphs

We give a simple, explicit, sufficient condition for the existence of a sector of minimal growth for second order regular singular differential operators on graphs. We specifically consider operators with a singular potential of Coulomb type and base our analysis on the theory of elliptic cone operators.

math.AP↗

arXiv2006

On cyclic numbers and an extension of Midy's theorem

In this note we consider fractions of the form 1/m and their floating-point representation in various arithmetic bases. For instance, what is 1/7 in base 2005? And, what about 1/4? We give a simple algorithm to answer these questions. In addition, we discuss an extension of Midy's theorem whose proof relies on elementary modular arithmetic.

math.NT↗