The Number of Chains of Subgroups in the Lattice of Subgroups of the Dicyclic Group

The present theory of finite reservoirs is not rich in general theorems even when of the simple Moran type, with unit draft and stationary discrete independent-sequence inflows. For the corresponding systems with unbounded capacity however there are two classes of results which have been known for some time. One of these classes is concerned with the time-dependent solution, where the theory provides a functional equation for the generating function of the time to first emptiness (Kendall (1957)), and the other with the asymptotic stationary distribution of reservoir contents, for which an explicit formula for the generating function is available (Moran (1959)).

Explicit formula of a new class of q-Hermite-based Apostol-type polynomials and generalization

Notes on Number Theory and Discrete Mathematics ◽

10.7546/nntdm.2020.26.4.93-102 ◽

2020 ◽

Vol 26 (4) ◽

pp. 93-102

Author(s):

Mouloud Goubi ◽

Keyword(s):

Explicit Formula ◽

Present Article ◽

Generating Function ◽

New Class

The present article deals with a recent study of a new class of q-Hermite-based Apostol-type polynomials introduced by Waseem A. Khan and Divesh Srivastava. We give their explicit formula and study a generalized class depending in any q-analog generating function.

PARTITIONS WITH AN ARBITRARY NUMBER OF SPECIFIED DISTANCES

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972719000650 ◽

2019 ◽

Vol 101 (1) ◽

pp. 35-39 ◽

Cited By ~ 1

Author(s):

BERNARD L. S. LIN

Keyword(s):

Explicit Formula ◽

Generating Function ◽

Generating Functions ◽

Arbitrary Number ◽

Positive Integers

For positive integers $t_{1},\ldots ,t_{k}$, let $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ (respectively $p(n,t_{1},t_{2},\ldots ,t_{k})$) be the number of partitions of $n$ such that, if $m$ is the smallest part, then each of $m+t_{1},m+t_{1}+t_{2},\ldots ,m+t_{1}+t_{2}+\cdots +t_{k-1}$ appears as a part and the largest part is at most (respectively equal to) $m+t_{1}+t_{2}+\cdots +t_{k}$. Andrews et al. [‘Partitions with fixed differences between largest and smallest parts’, Proc. Amer. Math. Soc.143 (2015), 4283–4289] found an explicit formula for the generating function of $p(n,t_{1},t_{2},\ldots ,t_{k})$. We establish a $q$-series identity from which the formulae for the generating functions of $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ and $p(n,t_{1},t_{2},\ldots ,t_{k})$ can be obtained.

INTERSECTION THEORY ON SYMPLECTIC QUOTIENTS OF PRODUCTS OF SPHERES

International Journal of Mathematics ◽

10.1142/s0129167x01000678 ◽

2001 ◽

Vol 12 (01) ◽

pp. 97-111 ◽

Cited By ~ 1

Author(s):

TATSURU TAKAKURA

Keyword(s):

Explicit Formula ◽

Generating Function ◽

Cohomology Ring ◽

Geometric Quantization ◽

Intersection Theory ◽

Symplectic Reduction ◽

Poincare Duality ◽

Diagonal Action ◽

Symplectic Quotients ◽

Products Of Spheres

We present an explicit formula for cohomology intersection pairings on an arbitrary smooth symplectic quotient of products of 2-spheres, by the standard diagonal action of SO3, without using known results on relations in the cohomology ring. By the Poincaré duality, it contains all the information enough to recover the structure of the cohomology ring. Our method is based on the commutativity of geometric quantization and symplectic reduction, originating from a conjecture of Guillemin-Sternberg. In fact, it enables us to derive a formula for the generating function of the intersection pairings.

Euler–Catalan’s Number Triangle and Its Application

Symmetry ◽

10.3390/sym12040600 ◽

2020 ◽

Vol 12 (4) ◽

pp. 600 ◽

Cited By ~ 1

Author(s):

Yuriy Shablya ◽

Dmitry Kruchinin

Keyword(s):

Explicit Formula ◽

Generating Function ◽

Binary Trees ◽

Combinatorial Objects ◽

Left Branch ◽

Bivariate Generating Function

In this paper, we study such combinatorial objects as labeled binary trees of size n with m ascents on the left branch and labeled Dyck n-paths with m ascents on return steps. For these combinatorial objects, we present the relation of the generated number triangle to Catalan’s and Euler’s triangles. On the basis of properties of Catalan’s and Euler’s triangles, we obtain an explicit formula that counts the total number of such combinatorial objects and a bivariate generating function. Combining the properties of these two number triangles allows us to obtain different combinatorial objects that may have a symmetry, for example, in their form or in their formulas.

Counting Stirling permutations by number of pushes

Pure Mathematics and Applications ◽

10.1515/puma-2015-0038 ◽

2020 ◽

Vol 29 (1) ◽

pp. 17-27

Author(s):

Toufik Mansour ◽

Mark Shattuck

Keyword(s):

Explicit Formula ◽

Generating Function ◽

Joint Distribution ◽

Inverse Matrix ◽

Infinite Matrix ◽

Do So

AbstractLet 𝒯(k)n denote the set of k-Stirling permutations having n distinct letters. Here, we consider the number of steps required (i.e., pushes) to rearrange the letters of a member of 𝒯(k)n so that they occur in non-decreasing order. We find recurrences for the joint distribution on 𝒯(k)n for the statistics recording the number of levels (i.e., occurrences of equal adjacent letters) and pushes. When k = 2, an explicit formula for the ordinary generating function of this distribution is also found. In order to do so, we determine the LU-decomposition of a certain infinite matrix having polynomial entries which enables one to compute explicitly the inverse matrix.

Generalized Gaussian Jacobsthal and Generalized Gaussian Jacobsthal Lucas Polynomial

International Journal of Engineering and Advanced Technology - Regular Issue ◽

10.35940/ijeat.f1130.0986s319 ◽

2019 ◽

Vol 8 (6S3) ◽

pp. 708-712

Keyword(s):

Explicit Formula ◽

Generating Function ◽

Binet Formula ◽

Q Matrix

The aim of the present paper is to present a generalization of Gaussian Jacobsthal polynomial and Gaussian Jacobsthal Lucas polynomial. Present paper extends the work of Asci and Gurel [3]. Some important generalizations of Generating function, Binet formula, Explicit formula, Q matrix and determinantal representations of these polynomials are also produced.

Counting self-dual interval orders

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2932 ◽

2011 ◽

Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽

Author(s):

Vít Jelínek

Keyword(s):

Explicit Formula ◽

Generating Function ◽

Interval Orders ◽

Maximal Elements ◽

Triangular Matrices ◽

Bijective Correspondence ◽

Upper Triangular Matrices ◽

Positive Entry ◽

International Audience ◽

Natural Statistics

International audience In this paper, we first derive an explicit formula for the generating function that counts unlabeled interval orders (a.k.a. (2+2)-free posets) with respect to several natural statistics, including their size, magnitude, and the number of minimal and maximal elements. In the second part of the paper, we derive a generating function for the number of self-dual unlabeled interval orders, with respect to the same statistics. Our method is based on a bijective correspondence between interval orders and upper-triangular matrices in which each row and column has a positive entry. Dans cet article, on obtient une expression explicite pour la fonction génératrice du nombre des ensembles partiellement ordonnés (posets) qui évitent le motif (2+2). La fonction compte ces ensembles par rapport à plusieurs statistiques naturelles, incluant le nombre d'éléments, le nombre de niveaux, et le nombre d'éléments minimaux et maximaux. Dans la deuxième partie, on obtient une expression similaire pour la fonction génératrice des posets autoduaux évitant le motif (2+2). On obtient ces résultats à l'aide d'une bijection entre les posets évitant (2+2) et les matrices triangulaires supérieures dont chaque ligne et chaque colonne contient un élément positif.

A multi-parameter generalization of the symmetric algorithm

Mathematica Slovaca ◽

10.1515/ms-2017-0137 ◽

2018 ◽

Vol 68 (4) ◽

pp. 699-712

Author(s):

José L. Ramírez ◽

Mark Shattuck

Keyword(s):

Explicit Formula ◽

Generating Function ◽

Fibonacci Numbers ◽

Recursive Formula ◽

Binomial Coefficient ◽

Fibonacci Polynomials ◽

Combinatorial Argument ◽

Recurrent Sequences ◽

Seidel Method ◽

Multivariate Generalization

Abstract The symmetric algorithm is a variant of the well-known Euler-Seidel method which has proven useful in the study of linearly recurrent sequences. In this paper, we introduce a multivariate generalization of the symmetric algorithm which reduces to it when all parameters are unity. We derive a general explicit formula via a combinatorial argument and also an expression for the row generating function. Several applications of our algorithm to the q-Fibonacci and q-hyper-Fibonacci numbers are discussed. Among our results is an apparently new recursive formula for the Carlitz Fibonacci polynomials. Finally, a (p, q)-analogue of the algorithm is introduced and an explicit formula for it in terms of the (p, q)-binomial coefficient is found.

A note on the time-dependent and the stationary behaviour of a semi-infinite reservoir subject to a combination of Markovian inflows

Journal of Applied Probability ◽

10.2307/3212234 ◽

1971 ◽

Vol 8 (4) ◽

pp. 708-715 ◽

Cited By ~ 3

Author(s):

Emlyn H. Lloyd

Keyword(s):

Functional Equation ◽

Explicit Formula ◽

Generating Function ◽

Present Theory ◽

The Other ◽

Time Dependent ◽

Independent Sequence ◽

Time Dependent Solution ◽

Infinite Reservoir ◽

Stationary Behaviour

The present theory of finite reservoirs is not rich in general theorems even when of the simple Moran type, with unit draft and stationary discrete independent-sequence inflows. For the corresponding systems with unbounded capacity however there are two classes of results which have been known for some time. One of these classes is concerned with the time-dependent solution, where the theory provides a functional equation for the generating function of the time to first emptiness (Kendall (1957)), and the other with the asymptotic stationary distribution of reservoir contents, for which an explicit formula for the generating function is available (Moran (1959)).