Genocchi Numbers Research Articles

Terrain-like graphs and the median Genocchi numbers

A graph with vertex set {1,…,n} is terrain-like if, for any edge pair {a,c}, {b,d} with a<b<c<d, the edge {a,d} also exists. Terrain-like graphs frequently appear in geometry in the context of visibility graphs. We show that terrain-like graphs are counted by the median Genocchi numbers. To this end, we show a natural bijection between terrain-like graphs and Dumont derangements of the second kind, with the graph’s edges representing individual transpositions.

Cycles of even-odd drop permutations and continued fractions of Genocchi numbers

Recently Lazar and Wachs proved two new permutation models, called D-permutations and E-permutations, for Genocchi and median Genocchi numbers. In a follow-up, Eu et al. studied the even-odd descent permutations, which are in bijection with E-permutations. We generalize Eu et al.'s descent polynomials with eight statistics and obtain an explicit J-fraction formula for their ordinary generaing function. The J-fraction permits us to confirm two conjectures of Lazar-Wachs about cycles of D and E permutations and obtain a (p,q)-analogue of Eu et al.'s gamma-formula. Moreover, the (p,q) gamma-coefficients have the same factorization flavor as the gamma-coefficients of Brändén's (p,q)-Eulerian polynomials.

A Note on Higher Order Degenerate Changhee–Genocchi Numbers and Polynomials of the Second Kind

In this paper, we consider the higher order degenerate Changhee–Genocchi polynomials of the second kind by using generating functions and the Riordan matrix methods. At the same time, we give some properties of the higher order degenerate Changhee–Genocchi polynomials of the second kind. In addition, we establish some new equalities involving the higher order degenerate Changhee–Genocchi polynomials of the second kind, the generalized Bell polynomials, higher order Changhee polynomials, the higher order degenerate Daehee polynomials of the second kind, Lah numbers and Stirling numbers, etc.

The integrality of the Genocchi numbers obtained through a new identity and other results

In this note, we investigate some properties of the integer sequence of general term a_n := \sum_{k = 0}^{n - 1} k! (n - k - 1)! (\forall n \geq 1) to derive a new identity of the Genocchi numbers G_n (n \in \mathbb{N}), which immediately shows that G_n \in \mathbb{Z} for any n \in \mathbb{N}. In another direction, we obtain nontrivial lower bounds for the 2-adic valuations of the rational numbers \sum_{k = 1}^{n} \frac{2^k}{k}.

New Type of Degenerate Changhee–Genocchi Polynomials

A remarkably large number of polynomials and their extensions have been presented and studied. In this paper, we consider a new type of degenerate Changhee–Genocchi numbers and polynomials which are different from those previously introduced by Kim. We investigate some properties of these numbers and polynomials. We also introduce a higher-order new type of degenerate Changhee–Genocchi numbers and polynomials which can be represented in terms of the degenerate logarithm function. Finally, we derive their summation formulae.

Gamma-Positivity for a Refinement of Median Genocchi Numbers

We study the generating function of descent numbers for the permutations with descent pairs of prescribed parities, the distribution of which turns out to be a refinement of median Genocchi numbers. We prove the $\gamma$-positivity for the polynomial and derive the generating function for the $\gamma$-vectors, expressed in the form of continued fraction. We also come up with an artificial statistic that gives a $q$-analogue of the $\gamma$-positivity for the permutations with descents only allowed from an odd value to an odd value.

The homogenized Linial arrangement and Genocchi numbers

We study the intersection lattice of a hyperplane arrangement recently introduced by Hetyei who showed that the number of regions of the arrangement is a median Genocchi number. Using a different method, we refine Hetyei's result by providing a combinatorial interpretation of the coefficients of the characteristic polynomial of the intersection lattice of this arrangement. The Genocchi numbers count a class of permutations known as Dumont permutations and the median Genocchi numbers count the derangements in this class. We show that the signless coefficients of the characteristic polynomial count Dumont-like permutations with a given number of cycles. This enables us to derive formulas for the generating function of the characteristic polynomial, which reduce to known formulas for the generating functions of the Genocchi numbers and the median Genocchi numbers. As a byproduct of our work, we obtain new models for the Genocchi and median Genocchi numbers.Mathematics Subject Classifications: 52C35 (Primary), 05A05, 05A15, 05B35, 06A07, 11B68 (Secondary)

On the Okounkov-Olshanski formula for standard tableaux of skew shapes

The classical hook length formula counts the number of standard tableaux of straight shapes. In 1996, Okounkov and Olshanski found a positive formula for the number of standard Young tableaux of a skew shape. We prove various properties of this formula, including three determinantal formulas for the number of nonzero terms, an equivalence between the Okounkov-Olshanski formula and another skew tableaux formula involving Knutson-Tao puzzles, and two $q$-analogues for reverse plane partitions, which complement work by Stanley and Chen for semistandard tableaux. We also give several reformulations of the formula, including two in terms of the excited diagrams appearing in a more recent skew tableaux formula by Naruse. Lastly, for thick zigzag shapes we show that the number of nonzero terms is given by a determinant of the Genocchi numbers and improve on known upper bounds by Morales-Pak-Panova on the number of standard tableaux of these shapes.Mathematics Subject Classifications: 05A15, 05A15, 05A19, 05A05

A Family of Generalized Legendre-Based Apostol-Type Polynomials

Numerous polynomials, their extensions, and variations have been thoroughly explored, owing to their potential applications in a wide variety of research fields. The purpose of this work is to provide a unified family of Legendre-based generalized Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials, with appropriate constraints for the Maclaurin series. Then we look at the formulae and identities that are involved, including an integral formula, differential formulas, addition formulas, implicit summation formulas, and general symmetry identities. We also provide an explicit representation for these new polynomials. Due to the generality of the findings given here, various formulae and identities for relatively simple polynomials and numbers, such as generalized Bernoulli, Euler, and Genocchi numbers and polynomials, are indicated to be deducible. Furthermore, we employ the umbral calculus theory to offer some additional formulae for these new polynomials.

A new generalization of the Genocchi numbers and its consequence on the Bernoulli polynomials

This paper presents a new generalization of the Genocchi numbers and the Genocchi theorem. As consequences, we obtain some important families of integer-valued polynomials those are closely related to the Bernoulli polynomials. Denoting by ${(B_n)}_{n \in \mathbb{N}}$ the sequence of the Bernoulli numbers and by ${(B_n(X))}_{n \in \mathbb{N}}$ the sequence of the Bernoulli polynomials, we especially obtain that for any natural number $n$, the reciprocal polynomial of the polynomial $\big(B_n(X) - B_n\big)$ is integer-valued.

Asymptotic Approximations of Apostol-Genocchi Numbers and Polynomials

Asymptotic approximations of the Apostol-Genocchi numbers andpolynomials are derived using Fourier series and ordering of poles ofthe generating function. Asymptotic formulas for the Apostol-Eulernumbers and polynomials are obtained as consequence. Asymptoticformulas for special cases which include the Genocchi numbers andpolynomials are also explicitly stated.

Fubini polynomials and integer partitions

In this paper, we show that the geometric polynomials can be expressed as sums over integer partitions in two different ways. New formulas involving geometric numbers, Bernoulli numbers, and Genocchi numbers are derived in this context.

Sine and cosine types of generating functions

We introduce two sine and cosine types of generating functions in a general case and apply them to the generating functions of classical hypergeometric orthogonal polynomials as well as some widely investigated combinatorial numbers such as Bernoulli, Euler and Genocchi numbers. This approach can also be applied to other celebrated sequences.

P.013 The neural signature of confidence abnormalities in compulsive disorders: obsessive-compulsive disorder and gambling disorder

We show that Γ-free fillings and lonesum fillings of Ferrers shapes are equinumerous by applying a previously defined bijection on matrices for this more general case and by constructing a new bijection between Callan sequences and Dumont-like permutations. As an application, we give a new combinatorial interpretation of Genocchi numbers in terms of Callan sequences.Further, we recover some of Hetyei’s results on alternating acyclic tournaments. Finally, we present an interesting result in the case of certain other special shapes.

The expansion of a chord diagram and the Genocchi numbers

A chord diagram E is a set of chords of a circle such that no pair of chords has a common endvertex. Let v 1 , v 2 , …, v 2 n be a sequence of vertices arranged in clockwise order along a circumference. A chord diagram { v 1 v n + 1 , v 2 v n + 2 , …, v n v 2 n } is called an n -crossing and a chord diagram { v 1 v 2 , v 3 v 4 , …, v 2 n − 1 v 2 n } is called an n -necklace. For a chord diagram E having a 2 -crossing S = { x 1 x 3 , x 2 x 4 } , the expansion of E with respect to S is to replace E with E 1 = ( E \ S ) ∪ { x 2 x 3 , x 4 x 1 } or E 2 = ( E \ S ) ∪ { x 1 x 2 , x 3 x 4 } . Beginning from a given chord diagram E as the root, by iterating chord expansions in both ways, we have a binary tree whose all leaves are nonintersecting chord diagrams. Let NCD ( E ) be the multiset of the leaves. In this paper, the multiplicity of an n -necklace in NCD ( E ) is studied. Among other results, it is shown that the multiplicity of an n -necklace generated from an n -crossing equals the Genocchi number when n is odd and the median Genocchi number when n is even.

On Some Sequences of Polynomials Generating the Genocchi Numbers

We consider sequences of Genocchi numbers of the first and the second kind. For these numbers, an approach based on their representation using sequences of polynomials is developed. Based on this approach, for these numbers some identities generalizing the known identities are constructed.

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.

On Multi Poly-Genocchi Polynomials with Parameters a, b and c

Most identities of Genocchi numbers and polynomials are related to the well-known Benoulli and Euler polynomials. In this paper, multi poly-Genocchi polynomials with parameters a, b and c are defined by means of multiple parameters polylogarithm. Several properties of these polynomials are established including some recurrence relations and explicit formulas.

Lonesum and [formula omitted]-free [formula omitted]-[formula omitted] fillings of Ferrers shapes

We show that Γ-free fillings and lonesum fillings of Ferrers shapes are equinumerous by applying a previously defined bijection on matrices for this more general case and by constructing a new bijection between Callan sequences and Dumont-like permutations. As an application, we give a new combinatorial interpretation of Genocchi numbers in terms of Callan sequences.Further, we recover some of Hetyei’s results on alternating acyclic tournaments. Finally, we present an interesting result in the case of certain other special shapes.

A note on degenerate Genocchi and poly-Genocchi numbers and polynomials

Recently, Dolgy–Jang introduced the poly-Genocchi polynomials and numbers arising from the modified polyexponential function. In this paper, we study the degenerate poly-Genocchi polynomials and numbers constructed from the modified degenerate polyexponential function. We derive explicit expressions for those polynomials and numbers. Also, we obtain identities involving those polynomials and numbers and some other special numbers and polynomials. In addition, we investigate the higher-order degenerate Genocchi polynomials and find identities involving those polynomials and the higher-order Changhee polynomials.