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.

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.

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 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 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.

scholarly journals The Compensation Approach for Walks With Small Steps in the Quarter Plane

2013 ◽  
Vol 22 (2) ◽  
pp. 161-183 ◽  
Author(s):  
IVO J. B. F. ADAN ◽  
JOHAN S. H. van LEEUWAARDEN ◽  
KILIAN RASCHEL
Keyword(s):  
Probability Theory ◽  
Explicit Expression ◽  
Generating Function ◽  
General Class ◽  
Counting Problems ◽  
Asymptotic Expressions ◽  
Quarter Plane ◽  
Formula Group ◽  
Compensation Approach ◽  
Image Position

This paper is the first application of the compensation approach (a well-established theory in probability theory) to counting problems. We discuss how this method can be applied to a general class of walks in the quarter plane +2 with a step set that is a subset of \[ \{(-1,1),(-1,0),(-1,-1),(0,-1),(1,-1)\}\] in the interior of +2. We derive an explicit expression for the generating function which turns out to be non-holonomic, and which can be used to obtain exact and asymptotic expressions for the counting numbers.

scholarly journals Avoiding maximal parabolic subgroups of S_k

2000 ◽  
Vol Vol. 4 no. 1 ◽  
Author(s):  
Toufik Mansour ◽  
Alek Vainshtein
Keyword(s):  
Explicit Expression ◽  
Generating Function ◽  
Parabolic Subgroups ◽  
International Audience ◽  
Maximal Parabolic Subgroups

International audience We find an explicit expression for the generating function of the number of permutations in S_n avoiding a subgroup of S_k generated by all but one simple transpositions. The generating function turns out to be rational, and its denominator is a rook polynomial for a rectangular board.

scholarly journals $321$-Polygon-Avoiding Permutations and Chebyshev Polynomials

10.37236/1677 ◽  
2003 ◽  
Vol 9 (2) ◽  
Author(s):  
Toufik Mansour ◽  
Zvezdelina Stankova
Keyword(s):  
Symmetric Group ◽  
Explicit Expression ◽  
Generating Function ◽  
Chebyshev Polynomials ◽  
Schubert Varieties ◽  
Exact Enumeration ◽  
Linear Recurrences ◽  
Singular Loci ◽  
Constant Coefficients

A $321$-$k$-gon-avoiding permutation $\pi$ avoids $321$ and the following four patterns: $$k(k+2)(k+3)\cdots(2k-1)1(2k)23\cdots(k-1)(k+1),$$ $$k(k+2)(k+3)\cdots(2k-1)(2k)12\cdots(k-1)(k+1),$$ $$(k+1)(k+2)(k+3)\cdots(2k-1)1(2k)23\cdots k,$$ $$(k+1)(k+2)(k+3)\cdots(2k-1)(2k)123\cdots k.$$ The $321$-$4$-gon-avoiding permutations were introduced and studied by Billey and Warrington [BW] as a class of elements of the symmetric group whose Kazhdan-Lusztig, Poincaré polynomials, and the singular loci of whose Schubert varieties have fairly simple formulas and descriptions. Stankova and West [SW] gave an exact enumeration in terms of linear recurrences with constant coefficients for the cases $k=2,3,4$. In this paper, we extend these results by finding an explicit expression for the generating function for the number of $321$-$k$-gon-avoiding permutations on $n$ letters. The generating function is expressed via Chebyshev polynomials of the second kind.

scholarly journals Monadic Second-Order Classes of Forests with a Monadic Second-Order 0-1 Law

2012 ◽  
Vol Vol. 14 no. 1 (Automata, Logic and Semantics) ◽  
Author(s):  
Jason P. Bell ◽  
Stanley N. Burris ◽  
Karen A. Yeats
Keyword(s):  
Explicit Expression ◽  
Generating Function ◽  
Generating Functions ◽  
Second Order ◽  
Radius Of Convergence ◽  
Structure Theory ◽  
Strengthened Version ◽  
International Audience

Automata, Logic and Semantics International audience Let T be a monadic-second order class of finite trees, and let T(x) be its (ordinary) generating function, with radius of convergence rho. If rho >= 1 then T has an explicit specification (without using recursion) in terms of the operations of union, sum, stack, and the multiset operators n and (>= n). Using this, one has an explicit expression for T(x) in terms of the initial functions x and x . (1 - x(n))(-1), the operations of addition and multiplication, and the Polya exponentiation operators E-n, E-(>= n). Let F be a monadic-second order class of finite forests, and let F (x) = Sigma(n) integral(n)x(n) be its (ordinary) generating function. Suppose F is closed under extraction of component trees and sums of forests. Using the above-mentioned structure theory for the class T of trees in F, Compton's theory of 0-1 laws, and a significantly strengthened version of 2003 results of Bell and Burris on generating functions, we show that F has a monadic second-order 0-1 law iff the radius of convergence of F (x) is 1 iff the radius of convergence of T (x) is >= 1.

scholarly journals An urn model arising from all-optical networks

2006 ◽  
Vol 43 (4) ◽  
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 dj ≤ B balls are placed in distinct urns with uniform probability. Let Mi(j) denote the number of urns containing i balls at the end of stage j. An explicit expression for the joint factorial moments of M0(j) and M1(j) is obtained. A multivariate generating function for the joint factorial moments of Mi(j), 0 ≤ i ≤ I, is derived (where I is a positive integer). Finally, the case in which the dj, j ≥ 1, are independent, identically distributed random variables is investigated.

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