arXiv subjects

Guo-Niu Han

Publications and source records attributed to Guo-Niu Han.

At least 37 records · Page 2Linked to original sources

arXiv2020

On the roots of the Poupard and Kreweras polynomials

The Poupard polynomials are polynomials in one variable with integer coefficients, with some close relationship to Bernoulli and tangent numbers. They also have a combinatorial interpretation. We prove that every Poupard polynomial has all its roots on the unit circle. We also obtain the same property for another sequence of polynomials introduced by Kreweras and related to Genocchi numbers. This is obtained through a general statement about some linear operators acting on palindromic polynomials.

math.CO↗

arXiv2019

Hankel Continued fractions and Hankel determinants of the Euler numbers

The Euler numbers occur in the Taylor expansion of $\tan(x)+\sec(x)$. Since Stieltjes, continued fractions and Hankel determinants of the even Euler numbers, on the one hand, of the odd Euler numbers, on the other hand, have been widely studied separately. However, no Hankel determinants of the (mixed) Euler numbers have been obtained and explicitly calculated. The reason for that is that some Hankel determinants of the Euler numbers are null. This implies that the Jacobi continued fraction of the Euler numbers does not exist. In the present paper, this obstacle is bypassed by using the Hankel continued fraction, instead of the $J$-fraction. Consequently, an explicit formula for the Hankel determinants of the Euler numbers is being derived, as well as a full list of Hankel continued fractions and Hankel determinants involving Euler numbers. Finally, a new $q$-analog of the Euler numbers $E_n(q)$ based on our continued fraction is proposed. We obtain an explicit formula for $E_n(-1)$ and prove a conjecture by R. J. Mathar on these numbers.

math.CO↗

arXiv2019

On the existence of permutations conditioned by certain rational functions

We prove several conjectures made by Z.-W. Sun on the existence of permutations conditioned by certain rational functions. Furthermore, we fully characterize all integer values of the "inverse difference" rational function. Our proofs consist of both investigation of the mathematical properties of the rational functions and brute-force attack by computer for finding special permutations.

math.CO↗

arXiv2019

Skew doubled shifted plane partitions: calculus and asymptotics

Plane partitions have been widely studied in Mathematics since MacMahon. See, for example, the works by Andrews, Macdonald, Stanley, Sagan and Krattenthaler. The Schur process approach, introduced by Okounkov and Reshetikhin, and further developed by Borodin, Corwin, Corteel, Savelief and Vuletić, has been proved to be a powerful tool in the study of various kinds of plane partitions. The exact enumerations of ordinary plane partitions, shifted plane partitions and cylindric partitions could be derived from two summation formulas for Schur processes, namely, the open summation formula and the cylindric summation formula. In this paper, we establish a new summation formula for Schur processes, called the complete summation formula. As an application, we obtain the generating function and the asymptotic formula for the number of doubled shifted plane partitions, which can be viewed as plane partitions `shifted at the two sides'. We prove that the order of the asymptotic formula depends only on the diagonal width of the doubled shifted plane partition, not on the profile (the skew zone) itself. By using the same methods, the generating function and the asymptotic formula for the number of symmetric cylindric partitions are also derived.

math.CO↗

arXiv2018

Combinatorial proofs of some properties of tangent and Genocchi numbers

The tangent number $T_{2n+1}$ is equal to the number of increasing labelled complete binary trees with $2n+1$ vertices. This combinatorial interpretation immediately proves that $T_{2n+1}$ is divisible by $2^n$. However, a stronger divisibility property is known in the studies of Bernoulli and Genocchi numbers, namely, the divisibility of $(n+1)T_{2n+1}$ by $2^{2n}$. The traditional proofs of this fact need significant calculations. In the present paper, we provide a combinatorial proof of the latter divisibility by using the hook length formula for trees. Furthermore, our method is extended to $k$-ary trees, leading to a new generalization of the Genocchi numbers.

math.CO↗

arXiv2018

Difference operators for partitions and some applications

Motivated by the Nekrasov-Okounkov formula on hook lengths, the first author conjectured that the Plancherel average of the $2k$-th power sum of hook lengths of partitions with size $n$ is always a polynomial of $n$ for any $k\in \mathbb{N}$. This conjecture was generalized and proved by Stanley (Ramanujan J., 23(1--3): 91--105, 2010). In this paper, inspired by the work of Stanley and Olshanski on the differential poset of Young lattice, we study the properties of two kinds of difference operators $D$ and $D^-$ defined on functions of partitions. Even though the calculations for higher orders of $D$ are extremely complex, we prove that several well-known families of functions of partitions are annihilated by a power of the difference operator $D$. As an application, our results lead to several generalizations of classic results on partitions, including the marked hook formula, Stanley Theorem, Okada-Panova hook length formula, and Fujii-Kanno-Moriyama-Okada content formula. We insist that the Okada constants $K_r$ arise directly from the computation for a single partition $λ$, without the summation ranging over all partitions of size~$n$.

math.CO↗

arXiv2017

Polynomiality of certain average weights for oscillating tableaux

We prove that a family of average weights for oscillating tableaux are polynomials in two variables, namely, the length of the oscillating tableau and the size of the ending partition, which generalizes a result of Hopkins and Zhang. Several explicit and asymptotic formulas for the average weights are also derived.

math.CO↗

arXiv2017

Some useful theorems for asymptotic formulas and their applications to skew plane partitions and cylindric partitions

Inspired by the works of Dewar, Murty and Kotěšovec, we establish some useful theorems for asymptotic formulas. As an application, we obtain asymptotic formulas for the numbers of skew plane partitions and cylindric partitions. We prove that the order of the asymptotic formula for the skew plane partitions of fixed width depends only on the width of the region, not on the profile (the skew zone) itself, while this is not true for cylindric partitions.

math.CO↗

arXiv2017

Difference operators for partitions under the Littlewood decomposition

The concept of $t$-difference operator for functions of partitions is introduced to prove a generalization of Stanley's theorem on polynomiality of Plancherel averages of symmetric functions related to contents and hook lengths. Our extension uses a generalization of the notion of Plancherel measure, based on walks in the Young lattice with steps given by the addition of $t$-hooks. It is well-known that the hook lengths of multiples of $t$ can be characterized by the Littlewood decomposition. Our study gives some further information on the contents and hook lengths of other congruence classes modulo $t$.

math.CO↗

arXiv2016

New hook-content formulas for strict partitions

We introduce the difference operator for functions defined on strict partitions and prove a polynomiality property for a summation involving the hook length and content statistics. As an application, several new hook-content formulas for strict partitions are derived.

math.CO↗

arXiv2016

Computer assisted proof for Apwenian sequences related to Hankel determinants

An infinite $\pm 1$-sequence is called {\it Apwenian} if its Hankel determinant of order $n$ divided by $2^{n-1}$ is an odd number for every positive integer $n$. In 1998, Allouche, Peyrière, Wen and Wen discovered and proved that the Thue--Morse sequence is an Apwenian sequence by direct determinant manipulations. Recently, Bugeaud and Han re-proved the latter result by means of an appropriate combinatorial method. By significantly improving the combinatorial method, we prove that several other Apwenian sequences related to the Hankel determinants with Computer Assistance.

math.NT↗

arXiv2016

Andre Permutation Calculus; a Twin Seidel Matrix Sequence

Entringer numbers occur in the André permutation combinatorial set-up under several forms. This leads to the construction of a matrix-analog refinement of the tangent (resp. secant) numbers. Furthermore, closed expressions for the three-variate exponential generating functions for pairs of so-called Entringerian statistics are derived.

math.CO↗

arXiv2016

Polynomiality of some hook-content summations for doubled distinct and self-conjugate partitions

In 2009, the first author proved the Nekrasov-Okounkov formula on hook lengths for integer partitions by using an identity of Macdonald in the framework of type $\widetilde A$ affine root systems, and conjectured that some summations over the set of all partitions of size $n$ are always polynomials in $n$. This conjecture was generalized and proved by Stanley. Recently, Pétréolle derived two Nekrasov-Okounkov type formulas for $\widetilde C$ and $\widetilde C\,\check{}$ which involve doubled distinct and self-conjugate partitions. Inspired by all those previous works, we establish the polynomiality of some hook-content summations for doubled distinct and self-conjugate partitions.

math.CO↗

arXiv2015

Hankel determinants, Padé approximations, and irrationality exponents

The irrationality exponent of an irrational number $ξ$, which measures the approximation rate of $ξ$ by rationals, is in general extremely difficult to compute explicitly, unless we know the continued fraction expansion of $ξ$. Results obtained so far are rather fragmentary, and often treated case by case. In this work, we shall unify all the known results on the subject by showing that the irrationality exponents of large classes of automatic numbers and Mahler numbers (which are transcendental) are exactly equal to $2$. Our classes contain the Thue--Morse--Mahler numbers, the sum of the reciprocals of the Fermat numbers, the regular paperfolding numbers, which have been previously considered respectively by Bugeaud, Coons, and Guo, Wu and Wen, but also new classes such as the Stern numbers and so on. Among other ingredients, our proofs use results on Hankel determinants obtained recently by Han.

math.NT↗

arXiv2014

Flag statistics from the Ehrhart series of multi-hypersimplices

It is known that the normalized volume of standard hypersimplices (defined as some slices of the unit hypercube) are the Eulerian numbers. More generally, a recent conjecture of Stanley relates the Ehrhart series of hypersimplices with descents and excedences in permutations. This conjecture was proved by Nan Li, who also gave a generalization to colored permutations. In this article, we give another generalization to colored permutations, using the flag statistics introduced by Foata and Han. We obtain in particular a new proof of Stanley's conjecture, and some combinatorial identities relating pairs of Eulerian statistics on colored permutations.

math.CO↗

arXiv2014

Hankel Determinant Calculus for the Thue-Morse and related sequences

The Hankel determinants of certain automatic sequences $f$ are evaluated, based on a calculation modulo a prime number. In most cases, the Hankel determinants of automatic sequences do not have any closed-form expressions; the traditional methods, such as $LU$-decompo\-si\-tion and Jacobi continued fraction, cannot be applied directly. Our method is based on a simple idea: the Hankel determinants of each sequence $g$ equal to $f$ modulo $p$ are equal to the Hankel determinants of $f$ modulo $p$. The clue then consists of finding a nice sequence $g$, whose Hankel determinants have closed-form expressions. Several examples are presented, including a result saying that the Hankel determinants of the Thue-Morse sequence are nonzero, first proved by Allouche, Peyrière, Wen and Wen using determinant manipulation. The present approach shortens the proof of the latter result significantly. We also prove that the corresponding Hankel determinants do not vanish when the powers $2^n$ in the infinite product defining the $\pm 1$ Thue--Morse sequence are replaced by $3^n$.

math.CO↗

arXiv2014

A combinatorial proof of the non-vanishing of Hankel determinants of the Thue--Morse sequence

In 1998, Allouche, Peyrière, Wen and Wen established that the Hankel determinants associated with the Thue--Morse sequence on $\{-1, 1\}$ are always nonzero. Their proof depends on a set of sixteen recurrence relations. We present an alternative, purely combinatorial proof of the same result. We also re-prove a recent result of Coons on the non-vanishing of the Hankel determinants associated to two other classical integer sequences.

math.CO↗

arXiv2014

On $t$-extensions of the Hankel determinants of certain automatic sequences

In 1998, Allouche, Peyrière, Wen and Wen considered the Thue--Morse sequence, and proved that all the Hankel determinants of the period-doubling sequence are odd integral numbers. We speak of $t$-extension when the entries along the diagonal in the Hankel determinant are all multiplied by~$t$. Then we prove that the $t$-extension of each Hankel determinant of the period-doubling sequence is a polynomial in $t$, whose leading coefficient is the {\it only one} to be an odd integral number. Our proof makes use of the combinatorial set-up developed by Bugeaud and Han, which appears to be very suitable for this study, as the parameter $t$ counts the number of fixed points of a permutation. Finally, we prove that all the $t$-extensions of the Hankel determinants of the regular paperfolding sequence are polynomials in $t$ of degree less than or equal to $3$.

math.CO↗