scholarly journals Generating Functions Attached to some Infinite Matrices

10.37236/492 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Paul Monsky
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Infinite Matrix ◽  
Infinite Matrices ◽  
Positive Integers

Let $V$ be an infinite matrix with rows and columns indexed by the positive integers, and entries in a field $F$. Suppose that $v_{i,j}$ only depends on $i-j$ and is 0 for $|i-j|$ large. Then $V^{n}$ is defined for all $n$, and one has a "generating function" $G=\sum a_{1,1}(V^{n})z^{n}$. Ira Gessel has shown that $G$ is algebraic over $F(z)$. We extend his result, allowing $v_{i,j}$ for fixed $i-j$ to be eventually periodic in $i$ rather than constant. This result and some variants of it that we prove will have applications to Hilbert-Kunz theory.

PARTITIONS WITH AN ARBITRARY NUMBER OF SPECIFIED DISTANCES

2019 ◽  
Vol 101 (1) ◽  
pp. 35-39 ◽  
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.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

scholarly journals A General Family of q-Hypergeometric Polynomials and Associated Generating Functions

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1161
Author(s):  
Hari Mohan Srivastava ◽  
Sama Arjika
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Additional Parameter ◽  
Physical Sciences ◽  
General Family ◽  
Hypergeometric Polynomials

Basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and the basic (or q-) hypergeometric polynomials are studied extensively and widely due mainly to their potential for applications in many areas of mathematical and physical sciences. Here, in this paper, we introduce a general family of q-hypergeometric polynomials and investigate several q-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of q-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized q-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various q-results, which we investigate here, with those in several related earlier works on this subject. We conclude this paper by remarking that it will be a rather trivial and inconsequential exercise to give the so-called (p,q)-variations of the q-results, which we have investigated here, because the additional parameter p is obviously redundant.

scholarly journals Congruence successions in compositions

2014 ◽  
Vol Vol. 16 no. 1 (Combinatorics) ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck ◽  
Mark Wilson
Keyword(s):  
General Formula ◽  
Generating Function ◽  
Asymptotic Estimate ◽  
International Audience ◽  
Limiting Case ◽  
Positive Integers

Combinatorics International audience A composition is a sequence of positive integers, called parts, having a fixed sum. By an m-congruence succession, we will mean a pair of adjacent parts x and y within a composition such that x=y(modm). Here, we consider the problem of counting the compositions of size n according to the number of m-congruence successions, extending recent results concerning successions on subsets and permutations. A general formula is obtained, which reduces in the limiting case to the known generating function formula for the number of Carlitz compositions. Special attention is paid to the case m=2, where further enumerative results may be obtained by means of combinatorial arguments. Finally, an asymptotic estimate is provided for the number of compositions of size n having no m-congruence successions.

scholarly journals Generating Functions for Certain Weighted Cranks

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
Shreejit Bandyopadhyay ◽  
Ae Yee
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Colored Partitions ◽  
Unimodal Sequences

Recently, George Beck posed many interesting partition problems considering the number of ones in partitions. In this paper, we first consider the crank generating function weighted by the number of ones and obtain analytic formulas for this weighted crank function under conditions of the crank being less than or equal to some specific integer. We connect these cumulative and point crank functions to the generating functions of partitions with certain sizes of Durfee rectangles. We then consider a generalization of the crank for $k$-colored partitions, which was first introduced by Fu and Tang, and investigate the corresponding generating function for this crank weighted by the number of parts in the first subpartition of a $k$-colored partition. We show that the cumulative generating functions are the same as the generating functions for certain unimodal sequences.

scholarly journals Generating functions and congruences for 9-regular and 27-regular partitions in 3 colours

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
Nayandeep Deka Baruah ◽  
Hirakjyoti Das
Keyword(s):  
Generating Functions ◽  
Regular Partitions ◽  
Positive Integers

Let $b_{\ell;3}(n)$ denote the number of $\ell$-regular partitions of $n$ in 3 colours. In this paper, we find some general generating functions and new infinite families of congruences modulo arbitrary powers of $3$ when $\ell\in\{9,27\}$. For instance, for positive integers $n$ and $k$, we have\begin{align*}b_{9;3}\left(3^k\cdot n+3^k-1\right)&\equiv0~\left(\textup{mod}~3^{2k}\right),\\b_{27;3}\left(3^{2k+3}\cdot n+\dfrac{3^{2k+4}-13}{4}\right)&\equiv0~\left(\textup{mod}~3^{2k+5}\right).\end{align*}

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 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.

COUNTING NUMERICAL SEMIGROUPS WITH SHORT GENERATING FUNCTIONS

2011 ◽  
Vol 21 (07) ◽  
pp. 1217-1235 ◽  
Author(s):  
VÍCTOR BLANCO ◽  
PEDRO A. GARCÍA-SÁNCHEZ ◽  
JUSTO PUERTO
Keyword(s):  
Generating Function ◽  
Polynomial Time ◽  
Generating Functions ◽  
Time Complexity ◽  
Frobenius Number ◽  
Numerical Semigroups ◽  
Polynomial Time Complexity ◽  
Theoretical Results

This paper presents a new methodology to compute the number of numerical semigroups of given genus or Frobenius number. We apply generating function tools to the bounded polyhedron that classifies the semigroups with given genus (or Frobenius number) and multiplicity. First, we give theoretical results about the polynomial-time complexity of counting these semigroups. We also illustrate the methodology analyzing the cases of multiplicity 3 and 4 where some formulas for the number of numerical semigroups for any genus and Frobenius number are obtained.

A General Asymptotic Scheme for the Analysis of Partition Statistics

2014 ◽  
Vol 23 (6) ◽  
pp. 1057-1086 ◽  
Author(s):  
PETER J. GRABNER ◽  
ARNOLD KNOPFMACHER ◽  
STEPHAN WAGNER
Keyword(s):  
Generating Function ◽  
Asymptotic Expansions ◽  
Generating Functions ◽  
Singularity Analysis ◽  
Higher Moments ◽  
Integer Partitions ◽  
Classical Singularity ◽  
Random Integer ◽  
Partition Statistics ◽  
New Statistics

We consider statistical properties of random integer partitions. In order to compute means, variances and higher moments of various partition statistics, one often has to study generating functions of the form P(x)F(x), where P(x) is the generating function for the number of partitions. In this paper, we show how asymptotic expansions can be obtained in a quasi-automatic way from expansions of F(x) around x = 1, which parallels the classical singularity analysis of Flajolet and Odlyzko in many ways. Numerous examples from the literature, as well as some new statistics, are treated via this methodology. In addition, we show how to compute further terms in the asymptotic expansions of previously studied partition statistics.

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