scholarly journals Direct Bijective Computation of the Generating Series for 2 and 3-Connection Coefficients of the Symmetric Group

10.37236/3226 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Alejandro H. Morales ◽  
Ekaterina A. Vassilieva
Keyword(s):  
Symmetric Group ◽  
Closed Form ◽  
Main Ingredient ◽  
Irreducible Characters ◽  
Long Cycle ◽  
Connection Coefficients ◽  
Special Cases ◽  
Generating Series ◽  
Orientable Surfaces ◽  
Form Evaluation

We evaluate combinatorially certain connection coefficients of the symmetric group that count the number of factorizations of a long cycle as a product of three permutations. Such factorizations admit an important topological interpretation in terms of unicellular constellations on orientable surfaces. Algebraic computation of these coefficients was first done by Jackson using irreducible characters of the symmetric group. However, bijective computations of these coefficients are so far limited to very special cases. Thanks to a new bijection that refines the work of Schaeffer and Vassilieva, we give an explicit closed form evaluation of the generating series for these coefficients. The main ingredient in the bijection is a modified oriented tricolored tree tractable to enumerate. Finally, reducing this bijection to factorizations of a long cycle into two permutations, we get the analogue formula for the corresponding generating series.

scholarly journals Long Cycle Factorizations: Bijective Computation in the General Case

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Ekaterina A. Vassilieva
Keyword(s):  
Symmetric Group ◽  
Closed Form ◽  
Closed Form Expression ◽  
Form Expression ◽  
Cycle Structure ◽  
Irreducible Characters ◽  
Long Cycle ◽  
Number Of Factors ◽  
Generating Series ◽  
International Audience

International audience This paper is devoted to the computation of the number of ordered factorizations of a long cycle in the symmetric group where the number of factors is arbitrary and the cycle structure of the factors is given. Jackson (1988) derived the first closed form expression for the generating series of these numbers using the theory of the irreducible characters of the symmetric group. Thanks to a direct bijection we compute a similar formula and provide the first purely combinatorial evaluation of these generating series. Cet article est dédié au calcul du nombre de factorisations d’un long cycle du groupe symétrique pour lesquels le nombre de facteurs est arbitraire et la structure des cycles des facteurs est donnée. Jackson (1988) a dérivé la première expression compacte pour les séries génératrices de ces nombres en utilisant la théorie des caractères irréductibles du groupe symétrique. Grâce à une bijection directe nous démontrons une formule similaire et donnons ainsi la première évaluation purement combinatoire de ces séries génératrices.

scholarly journals On a family of logarithmic and exponential integrals occurring in probability and reliability theory

1994 ◽  
Vol 35 (4) ◽  
pp. 469-478 ◽  
Author(s):  
M. Aslam Chaudhry
Keyword(s):  
Closed Form ◽  
Recurrence Relation ◽  
Reliability Theory ◽  
Integral Function ◽  
Integral Representations ◽  
Exponential Integral ◽  
Error Functions ◽  
Special Cases ◽  
Image Position ◽  
Form Evaluation

AbstractWe define an integral function Iμ(α, x; a, b) for non-negative integral values of μ byIt is proved that Iμ(α, x; a, b) satisfies a functional recurrence relation which is exploited to find a closed form evaluation of some incomplete integrals. New integral representations of the exponential integral and complementary error functions are found as special cases.

scholarly journals Explicit generating series for connection coefficients

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Ekaterina A. Vassilieva
Keyword(s):  
Symmetric Functions ◽  
Double Coset ◽  
Type A ◽  
Explicit Computation ◽  
Zonal Polynomials ◽  
Connection Coefficients ◽  
Generating Series ◽  
International Audience ◽  
Commutative Subalgebras ◽  
Orientable Surfaces

International audience This paper is devoted to the explicit computation of generating series for the connection coefficients of two commutative subalgebras of the group algebra of the symmetric group, the class algebra and the double coset algebra. As shown by Hanlon, Stanley and Stembridge (1992), these series gives the spectral distribution of some random matrices that are of interest to statisticians. Morales and Vassilieva (2009, 2011) found explicit formulas for these generating series in terms of monomial symmetric functions by introducing a bijection between partitioned hypermaps on (locally) orientable surfaces and some decorated forests and trees. Thanks to purely algebraic means, we recover the formula for the class algebra and provide a new simpler formula for the double coset algebra. As a salient ingredient, we compute an explicit formulation for zonal polynomials indexed by partitions of type $[a,b,1^{n-a-b}]$. Cet article est dédié au calcul explicite des séries génératrices des constantes de structure de deux sous-algèbres commutatives de l'algèbre de groupe du groupe symétrique, l'algèbre de classes et l'algèbre de double classe latérale. Tel que montrè par Hanlon, Stanley and Stembridge (1992), ces séries déterminent la distribution spectrale de certaines matrices aléatoires importantes en statistique. Morales et Vassilieva (2009, 2011) ont trouvè des formules explicites pour ces séries génératrices en termes des monômes symétriques en introduisant une bijection entre les hypercartes partitionnées sur des surfaces (localement) orientables et certains arbres et forêts décorées. Grâce à des moyens purement algébriques, nous retrouvons la formule pour l'algèbre de classe et déterminons une nouvelle formule plus simple pour l'algèbre de double classe latérale. En tant que point saillant de notre démonstration nous calculons une formulation explicite pour les polynômes zonaux indexés par des partitions de type $[a,b,1^{n-a-b}]$.

On the integration of incomplete elliptic integrals

1994 ◽  
Vol 444 (1922) ◽  
pp. 525-532 ◽  
Keyword(s):  
Closed Form ◽  
Hypergeometric Functions ◽  
Elliptic Integrals ◽  
Fractional Integrals ◽  
Special Cases ◽  
Complete Elliptic Integrals ◽  
Incomplete Elliptic Integrals ◽  
Form Evaluation

We give a closed-form evaluation of Erdélyi-Kober fractional integrals, involving incomplete elliptic integrals of the first kind, F ( φ, k ), and of the second kind, E ( φ, k ), which are integrated either with respect to the modulus or the amplitude. This is made possible by representing F ( φ, k ) and E ( φ, k ) in terms of the Kampé de Fériet double hypergeometric functions. Reduction formulae for these enable us to simplify the solutions for thirteen special cases, including integrals involving complete elliptic integrals. The hypergeometric character of the incomplete integrals is useful for evaluations of other classes of integrals involving F ( φ, k ) and E ( φ, k ).

Closed form evaluation of sums containing squares of Fibonomial coefficients

2016 ◽  
Vol 66 (3) ◽  
Author(s):  
Emrah Kiliç ◽  
Helmut Prodinger
Keyword(s):  
Closed Form ◽  
Systematic Approach ◽  
Sums Of Squares ◽  
Lucas Numbers ◽  
Fibonomial Coefficients ◽  
Finite Products ◽  
Fibonacci And Lucas Numbers ◽  
Form Evaluation

AbstractWe give a systematic approach to compute certain sums of squares of Fibonomial coefficients with finite products of generalized Fibonacci and Lucas numbers as coefficients. The technique is to rewrite everything in terms of a variable

Thermal Stresses Due to Disturbance of Uniform Heat Flow by an Insulated Ovaloid Hole

1960 ◽  
Vol 27 (4) ◽  
pp. 635-639 ◽  
Author(s):  
A. L. Florence ◽  
J. N. Goodier
Keyword(s):  
Heat Flow ◽  
Closed Form ◽  
Thermal Stresses ◽  
Thermoelastic Problem ◽  
Uniform Heat ◽  
Special Cases

The linear thermoelastic problem is solved for a uniform heat flow disturbed by a hole of ovaloid form, which includes the ellipse and circle as special cases. Results for stress and displacement are found in closed form, by reducing the problem to one of boundary loading solvable by a method of Muskhelishvili.

scholarly journals A Note on a Triple Integral

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2056
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Closed Form ◽  
Closed Form Expression ◽  
Polynomial Functions ◽  
Fundamental Constants ◽  
Form Expression ◽  
Zero Distribution ◽  
Special Cases ◽  
Almost All

A closed form expression for a triple integral not previously considered is derived, in terms of the Lerch function. Almost all Lerch functions have an asymmetrical zero-distribution. The kernel of the integral involves the product of the logarithmic, exponential, quotient radical, and polynomial functions. Special cases are derived in terms of fundamental constants; results are summarized in a table. All results in this work are new.

scholarly journals On the Signatures of Ordered System Lifetimes

2014 ◽  
Vol 51 (1) ◽  
pp. 82-91 ◽  
Author(s):  
N. Balakrishnan ◽  
William Volterman
Keyword(s):  
Closed Form ◽  
Coherent Systems ◽  
Special Cases

The idea of the system signature is extended here to the case of ordered system lifetimes arising from a test of coherent systems with a signature. An expression is given for the computation of the ordered system signatures in terms of the usual system signature for system lifetimes. Some properties of the ordered system signatures are then established. Closed-form expressions for the ordered system signatures are obtained in some special cases, and some illustrative examples are presented.

On the theory of resonance integrals in the statistical region

1964 ◽  
Vol 1 (02) ◽  
pp. 335-346 ◽  
Author(s):  
A. Reichel ◽  
C. A. Wilkins
Keyword(s):  
Closed Form ◽  
Mean Value ◽  
Resonance Integral ◽  
Theoretical Terms ◽  
Special Cases ◽  
The Mean

The problem of determining infinitely dilute resonance integrals is formulated in renewal theoretical terms. The mean value of the integral for a single resonance is determined in simple closed form. On the assumption that Wigner's hypothesis holds, the resonance density is determined, and a usable approximation to it is derived. An expression for the infinitely dilute resonance integral in the statistical region is then given and its value calculated in special cases and compared with the results of a previous computation.

Fatigue Damage Accumulation Prediction for Combined Sine and Random Stresses

1988 ◽  
Vol 31 (3) ◽  
pp. 53-63
Author(s):  
Ronald Lambert
Keyword(s):  
Fatigue Life ◽  
Closed Form ◽  
Fatigue Damage ◽  
Damage Accumulation ◽  
Structural Elements ◽  
Numerical Examples ◽  
Stress Situation ◽  
Fatigue Damage Accumulation ◽  
Special Cases ◽  
Parameters Of Interest

Simple closed-form expressions have been derived to predict fatigue life, damage accumulation, and other fatigue parameters of interest for structural elements with combined sinusoidal (sine) and narrowband Gaussian random stresses. These equations are expressed in common engineering terms. The sine and random only stress situations are special cases of the more general combined sine/random stress situation. They also have application for establishing vibration workmanship screens. Numerical examples are also included.

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