hook length
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Author(s):  
Bernhard Heim ◽  
Markus Neuhauser

AbstractIn this paper we investigate growth properties and the zero distribution of polynomials attached to arithmetic functions g and h, where g is normalized, of moderate growth, and $$0<h(n) \le h(n+1)$$ 0 < h ( n ) ≤ h ( n + 1 ) . We put $$P_0^{g,h}(x)=1$$ P 0 g , h ( x ) = 1 and $$\begin{aligned} P_n^{g,h}(x) := \frac{x}{h(n)} \sum _{k=1}^{n} g(k) \, P_{n-k}^{g,h}(x). \end{aligned}$$ P n g , h ( x ) : = x h ( n ) ∑ k = 1 n g ( k ) P n - k g , h ( x ) . As an application we obtain the best known result on the domain of the non-vanishing of the Fourier coefficients of powers of the Dedekind $$\eta $$ η -function. Here, g is the sum of divisors and h the identity function. Kostant’s result on the representation of simple complex Lie algebras and Han’s results on the Nekrasov–Okounkov hook length formula are extended. The polynomials are related to reciprocals of Eisenstein series, Klein’s j-invariant, and Chebyshev polynomials of the second kind.


2021 ◽  
Vol 4 (1) ◽  
Author(s):  
Yumi Inoue ◽  
Miki Kinoshita ◽  
Mamoru Kida ◽  
Norihiro Takekawa ◽  
Keiichi Namba ◽  
...  

AbstractThe flagellar protein export apparatus switches substrate specificity from hook-type to filament-type upon hook assembly completion, thereby initiating filament assembly at the hook tip. The C-terminal cytoplasmic domain of FlhA (FlhAC) serves as a docking platform for flagellar chaperones in complex with their cognate filament-type substrates. Interactions of the flexible linker of FlhA (FlhAL) with its nearest FlhAC subunit in the FlhAC ring is required for the substrate specificity switching. To address how FlhAL brings the order to flagellar assembly, we analyzed the flhA(E351A/W354A/D356A) ΔflgM mutant and found that this triple mutation in FlhAL increased the secretion level of hook protein by 5-fold, thereby increasing hook length. The crystal structure of FlhAC(E351A/D356A) showed that FlhAL bound to the chaperone-binding site of its neighboring subunit. We propose that the interaction of FlhAL with the chaperon-binding site of FlhAC suppresses filament-type protein export and facilitates hook-type protein export during hook assembly.


2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Philippe Biane ◽  
Matthieu Josuat-Vergès

International audience It is known that the number of minimal factorizations of the long cycle in the symmetric group into a product of k cycles of given lengths has a very simple formula: it is nk−1 where n is the rank of the underlying symmetric group and k is the number of factors. In particular, this is nn−2 for transposition factorizations. The goal of this work is to prove a multivariate generalization of this result. As a byproduct, we get a multivariate analog of Postnikov's hook length formula for trees, and a refined enumeration of final chains of noncrossing partitions.


2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Guo-Niu Han ◽  
Huan Xiong

International audience We introduce the difference operator for functions defined on strict partitions and prove a polynomiality property for a summation involving the bar length (hook length) and content statistics. As an application, several new hook-content formulas for strict partitions are derived.


2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Alejandro H. Morales ◽  
Igor Pak ◽  
Greta Panova

International audience The celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give two q-analogues of Naruse's formula for the skew Schur functions and for counting reverse plane partitions of skew shapes. We also apply our results to border strip shapes and their generalizations.


2020 ◽  
Author(s):  
Alina Guse ◽  
Manfred Rohde ◽  
Marc Erhardt

AbstractHook-length control is a central checkpoint during assembly of the bacterial flagellum. During hook growth, a 405 amino acids (aa) protein, FliK, is intermittently secreted and thought to function as a molecular measuring tape that, in Salmonella, controls hook-length to 55 nm ± 6 nm. The underlying mechanism involves interactions of both the α-helical, N-terminal domain of FliK (FliKN) with the hook and hook cap, and of its C-terminal domain with a component of the export apparatus. However, various deletion mutants of FliKN display uncontrolled hook-length, which is not consistent with a ruler mechanism. Here, we carried out an extensive deletion analysis of FliKN to investigate its contribution in the hook-length control mechanism. We identified FliKN mutants deleted for up to 80 aa that retained wildtype motility. However, the short FliK variants did not produce shorter hook-lengths as expected from a physical ruler. Rather, the minimal length of the hook depends on the level of hook protein production and secretion. Our results thus support a model in which FliK functions as a hook growth terminator protein that limits the maximal length of the hook, and not as a molecular ruler that physically measures hook-length.


10.37236/8685 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Anthony Mendes ◽  
Sam Lindbloom-Airey

We prove a $q$-analogue of the modular hook length formula using position sequences. These position sequences, which correspond to moving the beads in a mathematical abacus, provide a new combinatorial interpretation for the characters of the irreducible representations of the symmetric group.


2019 ◽  
Vol 113 (4) ◽  
pp. 355-366 ◽  
Author(s):  
Bernhard Heim ◽  
Markus Neuhauser
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