scholarly journals A Closed Formula for the Number of Convex Permutominoes

10.37236/975 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Filippo Disanto ◽  
Andrea Frosini ◽  
Renzo Pinzani ◽  
Simone Rinaldi
Keyword(s):  
Generating Function ◽  
Functional Equations ◽  
Closed Formula ◽  
System Of Functional Equations

In this paper we determine a closed formula for the number of convex permutominoes of size $n$. We reach this goal by providing a recursive generation of all convex permutominoes of size $n+1$ from the objects of size $n$, according to the ECO method, and then translating this construction into a system of functional equations satisfied by the generating function of convex permutominoes. As a consequence we easily obtain also the enumeration of some classes of convex polyominoes, including stack and directed convex permutominoes.

scholarly journals On Walks Avoiding a Quadrant

10.37236/8019 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Kilian Raschel ◽  
Amélie Trotignon
Keyword(s):  
Generating Function ◽  
Functional Equations ◽  
Single Equation ◽  
Convex Cones ◽  
Symmetric Convex ◽  
System Of Functional Equations ◽  
Quarter Plane ◽  
Discrete Structures ◽  
Linear Transform ◽  
Shortest Queue

Two-dimensional (random) walks in cones are very natural both in combinatorics and probability theory: they are interesting for themselves and also because they are strongly related to other discrete structures. While walks restricted to the first quadrant have been studied a lot, the case of planar, non-convex cones – equivalent to the three-quarter plane after a linear transform – has been approached only recently. In this article we develop an analytic approach to the case of walks in three quadrants. The advantage of this method is to provide uniform treatment in the study of models corresponding to different step sets. After splitting the three quadrants in two symmetric convex cones, the method is composed of three main steps: write a system of functional equations satisfied by the counting generating function, which may be simplified into one single equation under symmetry conditions; transform the functional equation into a boundary value problem; and finally solve this problem, using a concept of anti-Tutte's invariant. The result is a contour-integral expression for the generating function. Such systems of functional equations also appear in queueing theory with the famous Join-the-Shortest-Queue model, which is still an open problem in the non-symmetric case.

scholarly journals Three-Letter-Pattern Avoiding Permutations and Functional Equations

10.37236/1077 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Ghassan Firro ◽  
Toufik Mansour
Keyword(s):  
Generating Function ◽  
Functional Equations ◽  
Recurrence Relations ◽  
System Of Functional Equations ◽  
Pattern Avoiding Permutations ◽  
Letter Pattern

We present an algorithm for finding a system of recurrence relations for the number of permutations of length $n$ that satisfy a certain set of conditions. A rewriting of these relations automatically gives a system of functional equations satisfied by the multivariate generating function that counts permutations by their length and the indices of the corresponding recurrence relations. We propose an approach to describing such equations. In several interesting cases the algorithm recovers and refines, in a unified way, results on $\tau$-avoiding permutations and permutations containing $\tau$ exactly once, where $\tau$ is any classical, generalized, and distanced pattern of length three.

scholarly journals A New Class of Higher-Order Hypergeometric Bernoulli Polynomials Associated with Lagrange–Hermite Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 648
Author(s):  
Ghulam Muhiuddin ◽  
Waseem Ahmad Khan ◽  
Ugur Duran ◽  
Deena Al-Kadi
Keyword(s):  
Generating Function ◽  
Functional Equations ◽  
Laguerre Polynomials ◽  
Hermite Polynomials ◽  
Bernoulli Polynomials ◽  
Higher Order ◽  
New Class

The purpose of this paper is to construct a unified generating function involving the families of the higher-order hypergeometric Bernoulli polynomials and Lagrange–Hermite polynomials. Using the generating function and their functional equations, we investigate some properties of these polynomials. Moreover, we derive several connected formulas and relations including the Miller–Lee polynomials, the Laguerre polynomials, and the Lagrange Hermite–Miller–Lee polynomials.

MEASURES OF LOCAL ENTROPIES FOR COMPOSITIVE FUZZY PARTITIONS

2011 ◽  
Vol 19 (04) ◽  
pp. 717-728 ◽  
Author(s):  
DORETTA VIVONA ◽  
MARIA DIVARI
Keyword(s):  
Functional Equations ◽  
Fuzzy Measure ◽  
System Of Functional Equations ◽  
Locality Property ◽  
Fuzzy Measures ◽  
Fuzzy Partitions

The aim of this paper is to characterize of the measures of entropies without probability or fuzzy measure for compositive fuzzy partitions, taking into account the so-called locality property. We propose a system of functional equations, whose solutions give some forms of entropies without probability or fuzzy measures.

scholarly journals Labels distance in bucket recursive trees with variable capacities of buckets

2021 ◽  
Vol 13 (2) ◽  
pp. 413-426
Author(s):  
S. Naderi ◽  
R. Kazemi ◽  
M. H. Behzadi
Keyword(s):  
Partial Differential Equations ◽  
Probability Distribution ◽  
Differential Equations ◽  
Generating Function ◽  
Generating Functions ◽  
Function Approach ◽  
Closed Formula ◽  
Tree Model ◽  
Partial Differential ◽  
Recursive Trees

Abstract The bucket recursive tree is a natural multivariate structure. In this paper, we apply a trivariate generating function approach for studying of the depth and distance quantities in this tree model with variable bucket capacities and give a closed formula for the probability distribution, the expectation and the variance. We show as j → ∞, lim-iting distributions are Gaussian. The results are obtained by presenting partial differential equations for moment generating functions and solving them.

scholarly journals Common Fixed Point Theorems in Metric Spaces with Applications

2018 ◽  
Vol 11 (4) ◽  
pp. 1177-1190
Author(s):  
Pushpendra Semwal
Keyword(s):  
Dynamic Programming ◽  
Fixed Point ◽  
Common Fixed Point ◽  
Existence And Uniqueness ◽  
Functional Equations ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
System Of Functional Equations ◽  
Contractive Type ◽  
Common Fixed Point Theorems

In this paper we investigate the existence and uniqueness of common fixed point theorems for certain contractive type of mappings. As an application the existence and uniqueness of common solutions for a system of functional equations arising in dynamic programming are discuss by using the our results.

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