scholarly journals Enumeration of Pin-Permutations

10.37236/544 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Frédérique Bassino ◽  
Mathilde Bouvel ◽  
Dominique Rossin
Keyword(s):  
Generating Function ◽  
Substitution Decomposition

In this paper, we study the class of pin-permutations, that is to say of permutations having a pin representation. This class has been recently introduced by Brignall, Huczynska and Vatter who used it to find properties (algebraicity of the generating function, decidability of membership) of classes of permutations, depending on the simple permutations this class contains. We give a recursive characterization of the substitution decomposition trees of pin-permutations, which allows us to compute the generating function of this class, and consequently to prove, as it is conjectured by Brignall, Ruškuc and Vatter, the rationality of this generating function. Moreover, we show that the basis of the pin-permutation class is infinite.

Characterization of convergence rates for the approximation of the stationary distribution of infinite monotone stochastic matrices

1996 ◽  
Vol 33 (04) ◽  
pp. 974-985 ◽  
Author(s):  
F. Simonot ◽  
Y. Q. Song
Keyword(s):  
Convergence Rate ◽  
Generating Function ◽  
Convergence Rates ◽  
Transition Probability ◽  
Stochastic Matrix ◽  
Input Sequence ◽  
Transition Probability Matrix ◽  
Stochastic Matrices ◽  
Preceding Result

Let P be an infinite irreducible stochastic matrix, recurrent positive and stochastically monotone and Pn be any n × n stochastic matrix with Pn ≧ Tn , where Tn denotes the n × n northwest corner truncation of P. These assumptions imply the existence of limit distributions π and π n for P and Pn respectively. We show that if the Markov chain with transition probability matrix P meets the further condition of geometric recurrence then the exact convergence rate of π n to π can be expressed in terms of the radius of convergence of the generating function of π. As an application of the preceding result, we deal with the random walk on a half line and prove that the assumption of geometric recurrence can be relaxed. We also show that if the i.i.d. input sequence (A(m)) is such that we can find a real number r 0 > 1 with , then the exact convergence rate of π n to π is characterized by r 0. Moreover, when the generating function of A is not defined for |z| > 1, we derive an upper bound for the distance between π n and π based on the moments of A.

scholarly journals The Maximum Independent Sets of de Bruijn Graphs of Diameter 3

10.37236/681 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Dustin A. Cartwright ◽  
María Angélica Cueto ◽  
Enrique A. Tobis
Keyword(s):  
Recurrence Relation ◽  
Generating Function ◽  
Independent Sets ◽  
De Bruijn Graph ◽  
Alphabet Size ◽  
De Bruijn Graphs ◽  
Exponential Generating Function ◽  
De Bruijn

The nodes of the de Bruijn graph $B(d,3)$ consist of all strings of length $3$, taken from an alphabet of size $d$, with edges between words which are distinct substrings of a word of length $4$. We give an inductive characterization of the maximum independent sets of the de Bruijn graphs $B(d,3)$ and for the de Bruijn graph of diameter three with loops removed, for arbitrary alphabet size. We derive a recurrence relation and an exponential generating function for their number. This recurrence allows us to construct exponentially many comma-free codes of length 3 with maximal cardinality.

scholarly journals RAMANUJAN CONGRUENCES FOR A CLASS OF ETA QUOTIENTS

2010 ◽  
Vol 06 (04) ◽  
pp. 835-847 ◽  
Author(s):  
JONAH SINICK
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Generating Functions ◽  
Complete Characterization ◽  
Ramanujan Congruences

We consider a class of generating functions analogous to the generating function of the partition function and establish a bound on the primes ℓ for which their coefficients c(n) obey congruences of the form c(ℓn + a) ≡ 0 ( mod ℓ). We apply this result to obtain a complete characterization of the congruences of the same form that the sequences cN(n) satisfy, where cN(n) is defined by [Formula: see text]. This last result answers a question of H.-C. Chan.

scholarly journals Induction Models on $\mathbb{N}$

10.29007/kvp3 ◽  
2020 ◽  
Author(s):  
A. Dileep ◽  
Kuldeep S. Meel ◽  
Ammar F. Sabili
Keyword(s):  
Computer Science ◽  
Generating Function ◽  
Generating Functions ◽  
Mathematical Induction ◽  
Base Case ◽  
And Mathematics ◽  
Definition Of ◽  
Induction Model

Mathematical induction is a fundamental tool in computer science and mathematics. Henkin [12] initiated the study of formalization of mathematical induction restricted to the setting when the base case B is set to singleton set containing 0 and a unary generating function S. The usage of mathematical induction often involves wider set of base cases and k−ary generating functions with different structural restrictions. While subsequent studies have shown several Induction Models to be equivalent, there does not exist precise logical characterization of reduction and equivalence among different Induction Models. In this paper, we generalize the definition of Induction Model and demonstrate existence and construction of S for given B and vice versa. We then provide a formal characterization of the reduction among different Induction Models that can allow proofs in one Induction Models to be expressed as proofs in another Induction Models. The notion of reduction allows us to capture equivalence among Induction Models.

scholarly journals Reliability Assessment Methodology for Massive Manufacturing Using Multi-Function Equipment

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
M. López-Campos ◽  
F. Kristjanpoller ◽  
P. Viveros ◽  
R. Pascual
Keyword(s):  
Reliability Analysis ◽  
Generating Function ◽  
Production System ◽  
Manufacturing Process ◽  
Textile Industry ◽  
Reliability Assessment ◽  
Operating Conditions ◽  
Failure Behavior ◽  
Assessment Methodology

Experience reveals that reliability varies depending on the characteristics of operation. The manufacturing process based on multifunction equipment gives a usual case of variation in operating conditions. This work presents a methodology for the reliability analysis of multifunction processes, using the RCM approach, and a modification of the Universal Generating Function (UGF) under a massive manufacturing context. The result is a characterization of reliability, for each piece of equipment and for the production system. The methodology is applied in a workshop of a textile industry, where there is prior evidence that the failure behavior varies according to the type of function executed by multifunction machines.

scholarly journals Exactly Solvable Balanced Tenable Urns with Random Entries via the Analytic Methodology

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Basile Morcrette ◽  
Hosam M. Mahmoud
Keyword(s):  
Differential Equations ◽  
Generating Function ◽  
Probability Generating Function ◽  
Isomorphism Theorem ◽  
Systems Of Differential Equations ◽  
Exactly Solvable ◽  
Exact Distributions ◽  
International Audience ◽  
Weighted Probability

International audience This paper develops an analytic theory for the study of some Pólya urns with random rules. The idea is to extend the isomorphism theorem in Flajolet et al. (2006), which connects deterministic balanced urns to a differential system for the generating function. The methodology is based upon adaptation of operators and use of a weighted probability generating function. Systems of differential equations are developed, and when they can be solved, they lead to characterization of the exact distributions underlying the urn evolution. We give a few illustrative examples.

Euler-Seidel method for certain combinatorial numbers and a new characterization of Fibonacci sequence

2009 ◽  
Vol 7 (2) ◽  
Author(s):  
István Mező ◽  
Ayhan Dil
Keyword(s):  
Generating Function ◽  
Fibonacci Sequence ◽  
Stirling Numbers ◽  
Second Order ◽  
Seidel Method ◽  
Exponential Generating Function ◽  
Hyperharmonic Numbers ◽  
Recurrence Sequences

AbstractIn this paper we use the Euler-Seidel method for deriving new identities for hyperharmonic and r-Stirling numbers. The exponential generating function is determined for hyperharmonic numbers, which result is a generalization of Gosper’s identity. A classification of second order recurrence sequences is also given with the help of this method.

scholarly journals Hereditary Semiorders and Enumeration of Semiorders by Dimension

10.37236/8140 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Mitchel T. Keller ◽  
Stephen J. Young
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Interval Order ◽  
Structural Description ◽  
Interval Orders ◽  
Structural Result ◽  
Forbidden Subposets ◽  
Open Question ◽  
Ascent Sequences

In 2010, Bousquet-Mélou et al. defined sequences of nonnegative integers called ascent sequences and showed that the ascent sequences of length $n$ are in one-to-one correspondence with the interval orders, i.e., the posets not containing the poset $\mathbf{2}+\mathbf{2}$. Through the use of generating functions, this provided an answer to the longstanding open question of enumerating the (unlabeled) interval orders. A semiorder is an interval order having a representation in which all intervals have the same length. In terms of forbidden subposets, the semiorders exclude $\mathbf{2}+\mathbf{2}$ and $\mathbf{1}+\mathbf{3}$. The number of unlabeled semiorders on $n$ points has long been known to be the $n$th Catalan number. However, describing the ascent sequences that correspond to the semiorders under the bijection of Bousquet-Mélou et al. has proved difficult. In this paper, we discuss a major part of the difficulty in this area: the ascent sequence corresponding to a semiorder may have an initial subsequence that corresponds to an interval order that is not a semiorder. We define the hereditary semiorders to be those corresponding to an ascent sequence for which every initial subsequence also corresponds to a semiorder. We provide a structural result that characterizes the hereditary semiorders and use this characterization to determine the ordinary generating function for hereditary semiorders. We also use our characterization of hereditary semiorders and the characterization of semiorders of dimension $3$ given by Rabinovitch to provide a structural description of the semiorders of dimension at most $2$. From this description, we are able to determine the ordinary generating function for the semiorders of dimension at most $2$.

Characterization of convergence rates for the approximation of the stationary distribution of infinite monotone stochastic matrices

1996 ◽  
Vol 33 (4) ◽  
pp. 974-985 ◽  
Author(s):  
F. Simonot ◽  
Y. Q. Song
Keyword(s):  
Convergence Rate ◽  
Generating Function ◽  
Convergence Rates ◽  
Transition Probability ◽  
Stochastic Matrix ◽  
Input Sequence ◽  
Transition Probability Matrix ◽  
Stochastic Matrices ◽  
Preceding Result

Let P be an infinite irreducible stochastic matrix, recurrent positive and stochastically monotone and Pn be any n × n stochastic matrix with Pn ≧ Tn, where Tn denotes the n × n northwest corner truncation of P. These assumptions imply the existence of limit distributions π and πn for P and Pn respectively. We show that if the Markov chain with transition probability matrix P meets the further condition of geometric recurrence then the exact convergence rate of πn to π can be expressed in terms of the radius of convergence of the generating function of π. As an application of the preceding result, we deal with the random walk on a half line and prove that the assumption of geometric recurrence can be relaxed. We also show that if the i.i.d. input sequence (A(m)) is such that we can find a real number r0 > 1 with , then the exact convergence rate of πn to π is characterized by r0. Moreover, when the generating function of A is not defined for |z| > 1, we derive an upper bound for the distance between πn and π based on the moments of A.

Morphological Characterization of Mycobacteriophage R1

1974 ◽  
Vol 32 ◽  
pp. 254-255
Author(s):  
B. L. Soloff ◽  
T. A. Rado
Keyword(s):  
Carbon Source ◽  
Stationary Phase ◽  
Growth Phase ◽  
Minimal Medium ◽  
Morphological Characterization ◽  
Logarithmic Growth Phase ◽  
Complete Lysis ◽  
Lysogenic Culture ◽  
Logarithmic Growth

Mycobacteriophage R1 was originally isolated from a lysogenic culture of M. butyricum. The virus was propagated on a leucine-requiring derivative of M. smegmatis, 607 leu−, isolated by nitrosoguanidine mutagenesis of typestrain ATCC 607. Growth was accomplished in a minimal medium containing glycerol and glucose as carbon source and enriched by the addition of 80 μg/ ml L-leucine. Bacteria in early logarithmic growth phase were infected with virus at a multiplicity of 5, and incubated with aeration for 8 hours. The partially lysed suspension was diluted 1:10 in growth medium and incubated for a further 8 hours. This permitted stationary phase cells to re-enter logarithmic growth and resulted in complete lysis of the culture.

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