scholarly journals Counting Finite Languages by Total Word Length

Integers ◽  
2011 ◽  
Vol 11 (6) ◽  
Author(s):  
Stefan Gerhold
Keyword(s):  
Explicit Expression ◽  
Generating Function ◽  
Word Length ◽  
Finite Alphabet ◽  
Alphabet Size ◽  
Total Length ◽  
Large Alphabet ◽  
Finite Language

AbstractWe investigate the number of sets of words that can be formed from a finite alphabet, counted by the total length of the words in the set. An explicit expression for the counting sequence is derived from the generating function, and asymptotics for large alphabet size and large total word length are discussed. Moreover, we derive a Gaussian limit law for the number of words in a random finite language.

An Enumeration Problem Related to the Number of Labelled Bi-Coloured Graphs

1961 ◽  
Vol 13 ◽  
pp. 217-220 ◽  
Author(s):  
C. Y. Lee
Keyword(s):  
Positive Integer ◽  
Explicit Expression ◽  
Generating Function ◽  
Finite Sets ◽  
Enumeration Problem ◽  
Ordered Pairs

We will consider the following enumeration problem. Let A and B be finite sets with α and β elements in each set respectively. Let n be some positive integer such that n ≦ αβ. A subset S of the product set A × B of exactly n distinct ordered pairs (ai, bj) is said to be admissible if given any a ∈ A and b ∈ B, there exist elements (ai, bj) and (ak, bl) (they may be the same) in S such that ai = a and bl = b. We shall find here a generating function for the number N(α, β n) of distinct admissible subsets of A × B and from this generating function, an explicit expression for N(α, β n). In obtaining this result, the idea of a cut probability is used. This approach in a problem of enumeration may be of interest.

scholarly journals A Newton Interpolation Approach to Generalized Stirling Numbers

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Aimin Xu
Keyword(s):  
Explicit Expression ◽  
Recurrence Relation ◽  
Generating Function ◽  
Stirling Numbers ◽  
Divided Differences ◽  
Newton Interpolation ◽  
Basic Properties ◽  
Generalized Stirling Numbers

We employ the generalized factorials to define a Stirling-type pair{s(n,k;α,β,r),S(n,k;α,β,r)}which unifies various Stirling-type numbers investigated by previous authors. We make use of the Newton interpolation and divided differences to obtain some basic properties of the generalized Stirling numbers including the recurrence relation, explicit expression, and generating function. The generalizations of the well-known Dobinski's formula are further investigated.

ON THE AVERAGE SIZE OF GLUSHKOV AND PARTIAL DERIVATIVE AUTOMATA

2012 ◽  
Vol 23 (05) ◽  
pp. 969-984 ◽  
Author(s):  
SABINE BRODA ◽  
ANTÓNIO MACHIAVELO ◽  
NELMA MOREIRA ◽  
ROGÉRIO REIS
Keyword(s):  
Partial Derivative ◽  
Upper Bound ◽  
Regular Expression ◽  
Regular Expressions ◽  
Average Case ◽  
Alphabet Size ◽  
Large Alphabet ◽  
Exact Counting ◽  
Average Transition ◽  
Average Size

In this paper, the relation between the Glushkov automaton [Formula: see text] and the partial derivative automaton [Formula: see text] of a given regular expression, in terms of transition complexity, is studied. The average transition complexity of [Formula: see text] was proved by Nicaud to be linear in the size of the corresponding expression. This result was obtained using an upper bound of the number of transitions of [Formula: see text]. Here we present a new quadratic construction of [Formula: see text] that leads to a more elegant and straightforward implementation, and that allows the exact counting of the number of transitions. Based on that, a better estimation of the average size is presented. Asymptotically, and as the alphabet size grows, the number of transitions per state is on average 2. Broda et al. computed an upper bound for the ratio of the number of states of [Formula: see text] to the number of states of [Formula: see text] which is about ½ for large alphabet sizes. Here we show how to obtain an upper bound for the number of transitions in [Formula: see text], which we then use to get an average case approximation. In conclusion, assymptotically, and for large alphabets, the size of [Formula: see text] is half the size of the [Formula: see text]. This is corroborated by some experiments, even for small alphabets and small regular expressions.

scholarly journals On Cherednik and Nazarov-Sklyanin large N limit construction for integrable many-body systems with elliptic dependence on momenta

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
A. Grekov ◽  
A. Zotov
Keyword(s):  
Explicit Expression ◽  
Generating Function ◽  
Infinite Number ◽  
Symmetric Functions ◽  
Many Body ◽  
Quantum Double ◽  
Large N ◽  
Elliptic Model ◽  
Many Body Systems ◽  
Number Of Particles

Abstract The infinite number of particles limit in the dual to elliptic Ruijsenaars model (coordinate trigonometric degeneration of quantum double elliptic model) is proposed using the Nazarov-Sklyanin approach. For this purpose we describe double-elliptization of the Cherednik construction. Namely, we derive explicit expression in terms of the Cherednik operators, which reduces to the generating function of Dell commuting Hamiltonians on the space of symmetric functions. Although the double elliptic Cherednik operators do not commute, they can be used for construction of the N → ∞ limit.

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 An urn model arising from all-optical networks

2006 ◽  
Vol 43 (04) ◽  
pp. 952-966
Author(s):  
John A. Morrison
Keyword(s):  
Positive Integer ◽  
Explicit Expression ◽  
Optical Networks ◽  
Generating Function ◽  
Random Variables ◽  
Urn Model ◽  
Factorial Moments ◽  
Occupancy Model ◽  
All Optical ◽  
Independent Identically Distributed

An occupancy model that has arisen in the investigation of randomized distributed schedules in all-optical networks is considered. The model consists of B initially empty urns, and at stage j of the process d j ≤ B balls are placed in distinct urns with uniform probability. Let M i (j) denote the number of urns containing i balls at the end of stage j. An explicit expression for the joint factorial moments of M 0(j) and M 1(j) is obtained. A multivariate generating function for the joint factorial moments of M i (j), 0 ≤ i ≤ I, is derived (where I is a positive integer). Finally, the case in which the d j , j ≥ 1, are independent, identically distributed random variables is investigated.

scholarly journals Restrictions and Generalizations on Comma-Free Codes

10.37236/114 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Alexander L. Churchill
Keyword(s):  
Word Length ◽  
Coding Theory ◽  
Correspondence Problem ◽  
Major Difficulty ◽  
Alphabet Size ◽  
New Class ◽  
Np Complete

A significant sector of coding theory is that of comma-free coding; that is, codes which can be received without the need of a letter used for word separation. The major difficulty is in finding bounds on the maximum number of comma-free words which can inhabit a dictionary. We introduce a new class called a self-reflective comma-free dictionary and prove a series of bounds on the size of such a dictionary based upon word length and alphabet size. We also introduce other new classes such as self-swappable comma-free codes and comma-free codes in q dimensions and prove preliminary bounds for these classes. Finally, we discuss the implications and applications of combining these original concepts, including their implications for the NP-complete Post Correspondence Problem.

scholarly journals Impulse Propagation in Compositions and Words

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Margaret Archibald ◽  
Aubrey Blecher ◽  
Charlotte Brennan ◽  
Arnold Knopfmacher ◽  
Toufik Mansour
Keyword(s):  
Generating Function ◽  
Finite Alphabet ◽  
Impulse Propagation ◽  
The Right

We consider compositions of n represented as bargraphs and subject these to repeated impulses which start from the left at the top level and destroy horizontally connected parts. This is repeated while moving to the right first and then downwards to the next row and the statistic of interest is the number of impulses needed to annihilate the whole composition. We achieve this by conceptualizing a generating function that tracks compositions as well as the number of impulses used. This conceptualization is repeated for words (over a finite alphabet) represented by bargraphs.

The generating function of amplitudes with N twisted and L untwisted states

2015 ◽  
Vol 30 (21) ◽  
pp. 1550121 ◽  
Author(s):  
Igor Pesando
Keyword(s):  
Green Function ◽  
Explicit Expression ◽  
Generating Function

We show that the generating function of all correlators with N twisted and L untwisted states, i.e. the Reggeon vertex for magnetized branes on [Formula: see text] can be computed once the correlator of N nonexcited twisted states and the corresponding Green function are known and we give an explicit expression as a functional of these objects.

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