scholarly journals New hook-content formulas for strict partitions

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Guo-Niu Han ◽  
Huan Xiong
Keyword(s):  
Difference Operator ◽  
Hook Length ◽  
International Audience ◽  
The Difference

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

scholarly journals A direct bijective proof of the hook-length formula

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Jean-Christophe Novelli ◽  
Igor Pak ◽  
Alexander V. Stoyanovskii
Keyword(s):  
Young Tableaux ◽  
Hook Length ◽  
Bijective Proof ◽  
International Audience ◽  
Standard Young Tableaux

International audience This paper presents a new proof of the hook-length formula, which computes the number of standard Young tableaux of a given shape. After recalling the basic definitions, we present two inverse algorithms giving the desired bijection. The next part of the paper presents the proof of the bijectivity of our construction. The paper concludes with some examples.

scholarly journals On the 2-adic order of Stirling numbers of the second kind and their differences

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Tamás Lengyel
Keyword(s):  
Positive Integer ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Binary Representation ◽  
Exact Order ◽  
Special Cases ◽  
International Audience ◽  
The Difference ◽  
Divisibility Properties ◽  
Positive Integers

International audience Let $n$ and $k$ be positive integers, $d(k)$ and $\nu_2(k)$ denote the number of ones in the binary representation of $k$ and the highest power of two dividing $k$, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that $\nu_2(S(2^n,k))=d(k)-1, 1\leq k \leq 2^n$. Here we prove that $\nu_2(S(c2^n,k))=d(k)-1, 1\leq k \leq 2^n$, for any positive integer $c$. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on $\nu_2(S(c2^{n+1}+u,k)-S(c2^n+u,k))$ for any nonnegative integer $u$, make a conjecture on the exact order and, for $u=0$, prove part of it when $k \leq 6$, or $k \geq 5$ and $d(k) \leq 2$. The proofs rely on congruential identities for power series and polynomials related to the Stirling numbers and Bell polynomials, and some divisibility properties.

scholarly journals Statistics on Lattice Walks and q-Lassalle Numbers

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Lenny Tevlin
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Integer Coefficients ◽  
Major Index ◽  
Lattice Walks ◽  
International Audience ◽  
The Difference ◽  
Positive Integers

International audience This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan. Cet document contient deux résultats. Tout d’abord, je vous propose un $q$-generalization d’une certaine séquence de nombres entiers positifs, liés à nombres de Catalan, introduites par Zeilberger (Lassalle, 2010). Ces $q$-integers sont des polynômes palindromiques à $q$ à coefficients entiers positifs. La positivité dépend de la positivité d’une certaine différence de produits de $q$-coefficients binomial.Pour ce faire, je vous présente une nouvelle inversion/major index sur les chemins du réseau. La différence de $q$-binomial coefficients est alors considérée comme une fonction de génération de trajets pondérés qui restent dans le demi-plan supérieur.

Periodic Points of the Difference Operator

1997 ◽  
Vol 28 (1) ◽  
pp. 20-26
Author(s):  
Chris Bernhardt ◽  
Thomas Yuster
Keyword(s):  
Difference Operator ◽  
Periodic Points ◽  
The Difference

scholarly journals Design of preview controller for a type of discrete-time interconnected systems

2020 ◽  
Vol 53 (3-4) ◽  
pp. 719-729
Author(s):  
Hao Xie ◽  
Fucheng Liao ◽  
Usman ◽  
Jiamei Deng
Keyword(s):  
Discrete Time ◽  
Difference Operator ◽  
Interconnected Systems ◽  
Preview Control ◽  
Interconnected System ◽  
Lyapunov Function Method ◽  
System Equations ◽  
Error System ◽  
The Difference ◽  
Theoretical Results

This article proposes and studies a problem of preview control for a type of discrete-time interconnected systems. First, adopting the technique of decentralized control, isolated subsystems are constructed by splitting the correlations between the systems. Utilizing the difference operator to the system equations and error vectors, error systems are built. Then, the preview controller is designed for the error system of each isolated subsystem. The controllers of error systems of isolated subsystems are aggregated as a controller of the interconnected system. Finally, by employing Lyapunov function method and the properties of non-singular M-matrix, the guarantee conditions for the existence of preview controllers for interconnected systems are given. The numerical simulation shows that the theoretical results are effective.

scholarly journals Dichotomy and positivity of neutral equations with nonautonomous past

2014 ◽  
Vol 8 (2) ◽  
pp. 224-242 ◽  
Author(s):  
Nguyen Huy ◽  
Pham Bang
Keyword(s):  
Exponential Dichotomy ◽  
Difference Operator ◽  
Functional Differential Equations ◽  
Sufficient Condition ◽  
Neutral Functional Differential Equations ◽  
Neutral Equations ◽  
Exponentially Stable ◽  
The Difference ◽  
Functional Differential ◽  
Delay Operator

Consider the linear partial neutral functional differential equations with nonautonomous past of the form (?/?t) F(u(t, ?)) = BFu(t, ?) + ?u(t, ?), t ? 0; (? / ?t) u(t, s) = (? / ?s) u(t, s) + A(s)u(t, s), t ? 0 ? s, where the function u(?, ?) takes values in a Banach space X. Under appropriate conditions on the difference operator F and the delay operator ? we prove that the solution semigroup for this system of equations is hyperbolic (or admits an exponential dichotomy) provided that the backward evolution family U = (U(t, s))t?s?0 generated by A(s) is uniformly exponentially stable and the operator B generates a hyperbolic semigroup (etB)t?0 on X. Furthermore, under the positivity conditions on (etB)t?0, U, F and ? we prove that the above-mentioned solution semigroup is positive and then show a sufficient condition for the exponential stability of this solution semigroup.

Difference Operators in Simulation Data Warehouses

2017 ◽  
Vol 12 (2) ◽  
pp. 347-354 ◽  
Author(s):  
Jing Zhao ◽  
◽  
Yoshiharu Ishikawa ◽  
Yukiko Wakita ◽  
Kento Sugiura
Keyword(s):  
Difference Operator ◽  
Difference Operators ◽  
Observation Data ◽  
Temporal Data ◽  
Data Warehouses ◽  
Simulation Data ◽  
Tsunami Disaster ◽  
Spatio Temporal ◽  
The Difference ◽  
Mass Evacuation

In analyzing observation data and simulation results, there are frequent demands for comparing more than one data on the same subject to detect any differences between them. For example, comparison of observation data for an object in a certain spatial domain at different times or comparison of spatial simulation data with different parameters. Therefore, this paper proposes the difference operator in spatio-temporal data warehouses, which store temporal and spatial observation data and simulation data. The requirements for the difference operator are summarized, and the approaches to implement them are presented. In addition, the proposed approach is applied to the mass evacuation of simulation data in a tsunami disaster, and its effectiveness is verified. Extensions of the difference operator and their applications are also discussed.

scholarly journals On the fine spectrum of the generalized difference operatorB(r,s)over the sequence spacesc0andc

2005 ◽  
Vol 2005 (18) ◽  
pp. 3005-3013 ◽  
Author(s):  
Bilâl Altay ◽  
Feyzı Başar
Keyword(s):  
Difference Operator ◽  
Sequence Spaces ◽  
Band Matrix ◽  
Fine Spectrum ◽  
Special Cases ◽  
The Difference ◽  
The Right

We determine the fine spectrum of the generalized difference operatorB(r,s)defined by a band matrix over the sequence spacesc0andc, and derive a Mercerian theorem. This generalizes our earlier work (2004) for the difference operatorΔ, and includes as other special cases the right shift and the Zweier matrices.

scholarly journals Minimal factorizations of a cycle: a multivariate generating function

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Philippe Biane ◽  
Matthieu Josuat-Vergès
Keyword(s):  
Symmetric Group ◽  
Generating Function ◽  
Simple Formula ◽  
Hook Length ◽  
Noncrossing Partitions ◽  
Long Cycle ◽  
Number Of Factors ◽  
Multivariate Generalization ◽  
International Audience

International audience It is known that the number of minimal factorizations of the long cycle in the symmetric group into a product of k cycles of given lengths has a very simple formula: it is nk−1 where n is the rank of the underlying symmetric group and k is the number of factors. In particular, this is nn−2 for transposition factorizations. The goal of this work is to prove a multivariate generalization of this result. As a byproduct, we get a multivariate analog of Postnikov's hook length formula for trees, and a refined enumeration of final chains of noncrossing partitions.

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