scholarly journals Asymptotics of Young Diagrams and Hook Numbers

10.37236/1307 ◽  
1997 ◽  
Vol 4 (1) ◽  
Author(s):  
Amitai Regev ◽  
Anatoly Vershik
Keyword(s):  
Asymptotic Theory ◽  
Asymptotic Methods ◽  
Symmetric Groups ◽  
Schur Functions ◽  
Young Diagrams ◽  
Arithmetic Properties ◽  
Hook Formula ◽  
Hook Numbers

Asymptotic calculations are applied to study the degrees of certain sequences of characters of symmetric groups. Starting with a given partition $\mu$, we deduce several skew diagrams which are related to $\mu$. To each such skew diagram there corresponds the product of its hook numbers. By asymptotic methods we obtain some unexpected arithmetic properties between these products. The authors do not know "finite", nonasymptotic proofs of these results. The problem appeared in the study of the hook formula for various kinds of Young diagrams. The proofs are based on properties of shifted Schur functions, due to Okounkov and Olshanski. The theory of these functions arose from the asymptotic theory of Vershik and Kerov of the representations of the symmetric groups.

scholarly journals Cut-and-join structure and integrability for spin Hurwitz numbers

2020 ◽  
Vol 80 (2) ◽  
Author(s):  
A. Mironov ◽  
A. Morozov ◽  
S. Natanzon
Keyword(s):  
Matrix Models ◽  
Generating Function ◽  
Schur Functions ◽  
Integrable Hierarchy ◽  
Young Diagrams ◽  
Hurwitz Numbers ◽  
Expansion Coefficients ◽  
Common Eigenfunctions

Abstract Spin Hurwitz numbers are related to characters of the Sergeev group, which are the expansion coefficients of the Q Schur functions, depending on odd times and on a subset of all Young diagrams. These characters involve two dual subsets: the odd partitions (OP) and the strict partitions (SP). The Q Schur functions $$Q_R$$QR with $$R\in \hbox {SP}$$R∈SP are common eigenfunctions of cut-and-join operators $$W_\Delta $$WΔ with $$\Delta \in \hbox {OP}$$Δ∈OP. The eigenvalues of these operators are the generalized Sergeev characters, their algebra is isomorphic to the algebra of Q Schur functions. Similarly to the case of the ordinary Hurwitz numbers, the generating function of spin Hurwitz numbers is a $$\tau $$τ-function of an integrable hierarchy, that is, of the BKP type. At last, we discuss relations of the Sergeev characters with matrix models.

scholarly journals Characters of symmetric groups in terms of free cumulants and Frobenius coordinates

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Maciej Dolega ◽  
Valentin Féray ◽  
Piotr Sniady
Keyword(s):  
Young Diagram ◽  
Symmetric Groups ◽  
Main Tool ◽  
Combinatorial Formula ◽  
Young Diagrams ◽  
Free Cumulants ◽  
International Audience

International audience Free cumulants are nice and useful functionals of the shape of a Young diagram, in particular they give the asymptotics of normalized characters of symmetric groups $\mathfrak{S}(n)$ in the limit $n \to \infty$. We give an explicit combinatorial formula for normalized characters of the symmetric groups in terms of free cumulants. We also express characters in terms of Frobenius coordinates. Our formulas involve counting certain factorizations of a given permutation. The main tool are Stanley polynomials which give values of characters on multirectangular Young diagrams. Les cumulants libres sont des fonctions agréables et utiles sur l'ensemble des diagrammes de Young, en particulier, ils donnent le comportement asymptotiques des caractères normalisés du groupe symétrique $\mathfrak{S}(n)$ dans la limite $n \to \infty$. Nous donnons une formule combinatoire explicite pour les caractères normalisés du groupe symétrique en fonction des cumulants libres. Nous exprimons également les caractères en fonction des coordonnées de Frobenius. Nos formules font intervenir le nombre de certaines factorisations d'une permutation donnée. L'outil principal est la famille de polynômes de Stanley donnant les valeurs des caractères sur les diagrammes de Young multirectangulaires.

scholarly journals The Contribution to the Development of a Two-Dimensional Asymptotic Theory of the Three-Point Bending Behaviour of Multi-Layered Beams: Applications to Orthotropic Phase Sandwich Beams

2020 ◽  
Vol 5 (10) ◽  
pp. 1191-1198
Author(s):  
A. D. Pagui ◽  
A. E. Foudjet ◽  
J. S. T. Mabekou ◽  
T. R. S. N. Ekoume ◽  
P. K. Talla
Keyword(s):  
Asymptotic Method ◽  
Asymptotic Theory ◽  
Asymptotic Methods ◽  
Small Variation ◽  
Second Order ◽  
Sandwich Beams ◽  
Two Dimensional ◽  
Three Point Bending ◽  
Layered Beams ◽  
Bending Behaviour

The objective of this work is to present a methodology for analyzing the behavior in bending of the structure of sandwich beams base on the second order of asymptotic method. This work is in continuation with the work of Talla [1]. This work includes the knowledge of all the physical elastic constant of the sandwich beams. This result confirms the fact that the second order of asymptotic method doesn’t bring a significative change in the behavior of the solution until a certain point. The curves have been obtained by the software named python. This result was predictable because the asymptotic methods deal with small variation due to the presence of the epsilon parameter, which is very small.

scholarly journals Tableaux and plane partitions of truncated shapes (extended abstract)

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Greta Panova
Keyword(s):  
Generating Function ◽  
Schur Functions ◽  
Young Tableaux ◽  
Young Diagrams ◽  
Plane Partitions ◽  
Nous Trouvons ◽  
International Audience

International audience We consider a new kind of straight and shifted plane partitions/Young tableaux — ones whose diagrams are no longer of partition shape, but rather Young diagrams with boxes erased from their upper right ends. We find formulas for the number of standard tableaux in certain cases, namely a shifted staircase without the box in its upper right corner, i.e. truncated by a box, a rectangle truncated by a staircase and a rectangle truncated by a square minus a box. The proofs involve finding the generating function of the corresponding plane partitions using interpretations and formulas for sums of restricted Schur functions and their specializations. The number of standard tableaux is then found as a certain limit of this function. Nous considérons un nouveau type de partitions planes, ou de tableaux de Young, droits ou décalés, obtenus en privant leurs diagrammes de certaines cellules en haut à droite, et dans certains cas nous trouvons des formules d'énumération pour les tableaux standard. Les preuves impliquent le calcul de la fonction génératrice pour les partitions planes correspondantes, en utilisant des interprétations et des formules pour les sommes de fonctions de Schur restreintes et leurs spécialisations. Le nombre de tableaux standard est alors obtenu comme une certaine limite de cette fonction.

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