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Guo-Niu Han

Publications and source records attributed to Guo-Niu Han.

At least 55 records · Page 3Linked to original sources
arXiv2014

Hankel continued fraction and its applications

The Hankel determinants of a given power series $f$ can be evaluated by using the Jacobi continued fraction expansion of $f$. However the existence of the Jacobi continued fraction needs that all Hankel determinants of $f$ are nonzero. We introduce {\it Hankel continued fraction}, whose existene and unicity are guaranteed without any condition for the power series $f$. The Hankel determinants can also be evaluated by using the Hankel continued fraction. It is well known that the continued fraction expansion of a quadratic irrational number is ultimately periodic. We prove a similar result for power series. If a power series $f$ over a finite field satisfies a quadratic functional equation, then the Hankel continued fraction is ultimately periodic. As an application, we derive the Hankel determinants of several automatic sequences, in particular, the regular paperfolding sequence. Thus we provide an automatic proof of a result obtained by Guo, Wu and Wen, which was conjectured by Coons-Vrbik.

math.NT↗
arXiv2014

Explicit evaluations of the Hankel determinants of a Thue--Morse-like sequence

We obtain the explicit evaluations of the Hankel determinants of the formal power series $\prod_{k\geq 0}(1+Jx^{3^{k}})$ where $J={(\sqrt{-3}-1)}/2$, and prove that the sequence of Hankel determinants is an aperiodic automatic sequence taking value in $\{0, \pm 1, \pm J, \pm J^2\}$. This research is essentially inspired by the works about Hankel determinants of Thue--Morse-like sequences by Allouche, Peyrière, Wen and Wen (1998), Bacher (2006) and the first author (2013).

math.NT↗
arXiv2014

Efficient Legendre moment computation for grey level images

Legendre orthogonal moments have been widely used in the field of image analysis. Because their computation by a direct method is very time expensive, recent efforts have been devoted to the reduction of computational complexity. Nevertheless, the existing algorithms are mainly focused on binary images. We propose here a new fast method for computing the Legendre moments, which is not only suitable for binary images but also for grey levels. We first set up the recurrence formula of one-dimensional (1D) Legendre moments by using the recursive property of Legendre polynomials. As a result, the 1D Legendre moments of order p, Lp = Lp(0), can be expressed as a linear combination of Lp-1(1) and Lp-2(0). Based on this relationship, the 1D Legendre moments Lp(0) is thus obtained from the array of L1(a) and L0(a) where a is an integer number less than p. To further decrease the computation complexity, an algorithm, in which no multiplication is required, is used to compute these quantities. The method is then extended to the calculation of the two-dimensional Legendre moments Lpq. We show that the proposed method is more efficient than the direct method.

cs.CV↗
arXiv2014

Image reconstruction from limited range projections using orthogonal moments

A set of orthonormal polynomials is proposed for image reconstruction from projection data. The relationship between the projection moments and image moments is discussed in detail, and some interesting properties are demonstrated. Simulation results are provided to validate the method and to compare its performance with previous works.

cs.CV↗
arXiv2013

Finite Difference Calculus for Alternating Permutations

The finite difference equation system introduced by Christiane Poupard in the study of tangent trees is reinterpreted in the alternating permutation environment. It makes it possible to make a joint study of both tangent and secant trees and calculate the generating polynomial for alternating permutations by a new statistic, referred to as being the greater neighbor of the maximum.

math.CO↗
arXiv2013

Tree Calculus for Bivariable Difference Equations

Following Poupard's study of strictly ordered binary trees with respect to two parameters, namely, "end of minimal chain" and "parent of maximum leaf" a true Tree Calculus is being developed to solve a partial difference equation system and then make a joint study of those two statistics. Their joint distribution is shown to be symmetric and to be expressed in the form of an explicit three-variable generating function.

math.CO↗
arXiv2013

Secant Tree Calculus

A true Tree Calculus is being developed to make a joint study of the two statistics "eoc" (end of minimal chain) and "pom" (parent of maximum leaf) on the set of secant trees. Their joint distribution restricted to the set {eoc-pom<= 1} is shown to satisfy two partial difference equation systems, to be symmetric and to be expressed in the form of an explicit three-variable generating function.

math.CO↗
arXiv2013

Multivariable Tangent and Secant q-derivative Polynomials

The derivative polynomials introduced by Knuth and Buckholtz in their calculations of the tangent and secant numbers are extended to a multivariable $q$--environment. The $n$-th $q$-derivatives of the classical $q$-tangent and $q$-secant are each given two polynomial expressions. The first polynomial expression is indexed by triples of integers, the second by compositions of integers. The functional relation between those two classes is fully given by means of combinatorial techniques. Moreover, those polynomials are proved to be generating functions for so-called $t$-permutations by multivariable statistics. By giving special values to those polynomials we recover classical $q$-polynomials such as the Carlitz $q$-Eulerian polynomials and the $(t,q)$-tangent and -secant analogs recently introduced. They also provide $q$-analogs for the Springer numbers. Finally, the $t$-compositions used in this paper furnish a combinatorial interpretation to one of the Fibonacci triangles.

math.CO↗
arXiv2011

A multiset hook length formula and some applications

A multiset hook length formula for integer partitions is established by using combinatorial manipulation. As special cases, we rederive three hook length formulas, two of them obtained by Nekrasov-Okounkov, the third one by Iqbal, Nazir, Raza and Saleem, who have made use of the cyclic symmetry of the topological vertex. A multiset hook-content formula is also proved.

math.CO↗
arXiv2008

Hook lengths and shifted parts of partitions

Some conjectures on partition hook lengths, recently stated by the author, have been proved and generalized by Stanley, who also needed a formula by Andrews, Goulden and Jackson on symmetric functions to complete his derivation. Another identity on symmetric functions can be used instead. The purpose of this note is to prove it.

math.CO↗
arXiv2008

Hook lengths and 3-cores

Recently, the first author generalized a formula of Nekrasov and Okounkov which gives a combinatorial formula, in terms of hook lengths of partitions, for the coefficients of certain power series. In the course of this investigation, he conjectured that $A000731(n)=0$ if and only if $A033687(n)=0$. The numbers $A000731(n)$ are given in terms of hook lengths of partitions, while $A033687(n)$ equals the number of 3-core partitions of $n$. Here we prove this conjecture.

math.NT↗
arXiv2008

Discovering hook length formulas by expansion technique

We introduce the hook length expansion technique and explain how to discover old and new hook length formulas for partitions and plane trees. The new hook length formulas for trees obtained by our method can be proved rather easily, whereas those for partitions are much more difficult and some of them still remain open conjectures. We also develop a Maple package HookExp for computing the hook length expansion. The paper can be seen as a collection of hook length formulas for partitons and plane trees. All examples are illustrated by HookExp and, for many easy cases, expained by well-known combinatorial arguments.

math.CO↗
arXiv2008

An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths

We discover an explicit expansion formula for the powers $s$ of the Euler Product (or Dedekind $η$-function) in terms of hook lengths of partitions, where the exponent $s$ is any complex number. Several classical formulas have been derived for certain integers $s$ by Euler, Jacobi, Klein, Fricke, Atkin, Winquist, Dyson and Macdonald. In particular, Macdonald obtained expansion formulas for the integer exponents $s$ for which there exists a semi-simple Lie algebra of dimension $s$. For the type $A_l^{(a)}$ he has expressed the $(t^2-1)$-st power of the Euler Product as a sum of weighted integer vectors of length $t$ for any integer $t$. Kostant has considered the general case for any positive integer $s$ and obtained further properties. ----- The present paper proposes a new approach. We convert the weighted vectors of length $t$ used by Macdonald in his identity for type $A_l^{(a)}$ to weighted partitions with free parameter $t$, so that a new identity on the latter combinatorial structures can be derived without any restrictions on $t$. The surprise is that the weighted partitions have a very simple form in terms of hook lengths of partitions. As applications of our formula, we find some new identities about hook lengths, including the "marked hook formula". We also improve a result due to Kostant. The proof of the Main Theorem is based on Macdonald's identity for $A_l^{(a)}$ and on the properties of a bijection between $t$-cores and integer vectors constructed by Garvan, Kim and Stanton.

math.CO↗
arXiv2008

The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications

The paper is devoted to the derivation of the expansion formula for the powers of the Euler Product in terms of partition hook lengths, discovered by Nekrasov and Okounkov in their study of the Seiberg-Witten Theory. We provide a refinement based on a new property of t-cores, and give an elementary proof by using the Macdonald identities. We also obtain an extension by adding two more parameters, which appears to be a discrete interpolation between the Macdonald identities and the generating function for t-cores. Several applications are derived, including the "marked hook formula".

math.CO↗
arXiv2008

Yet another generalization of Postnikov's hook length formula for binary trees

We discover another one-parameter generalization of Postnikov's hook length formula for binary trees. The particularity of our formula is that the hook length $h_v$ appears as an exponent. As an application, we derive another simple hook length formula for binary trees when the underlying parameter takes the value 1/2.

math.CO↗
arXiv2007

Permutations with Extremal number of Fixed Points

We extend Stanley's work on alternating permutations with extremal number of fixed points in two directions: first, alternating permutations are replaced by permutations with a prescribed descent set; second, instead of simply counting permutations we study their generating polynomials by number of excedances. Several techniques are used: Desarmenien's desarrangement combinatorics, Gessel's hook-factorization and the analytical properties of two new permutation statistics "DEZ" and "lec". Explicit formulas for the maximal case are derived by using symmetric function tools.

math.CO↗
arXiv2007

Fix-Mahonian Calculus III; a Quadruple Distribution

A four-variable distribution on permutations is derived, with two dual combinatorial interpretations. The first one includes the number of fixed points "fix", the second the so-called "pix" statistic. This shows that the duality between derangements and desarrangements can be extended to the case of multivariable statistics. Several specializations are obtained, including the joint distribution of (des, exc), where "des" and "exc" stand for the number of descents and excedances, respectively.

math.CO↗