The Garsia–Haiman module is a bigraded Sn-module whose Frobenius image Macdonald polynomial. method of orbit harmonics promotes an Sn-set X to graded polynomial ring. can be applied prove cyclic sieving phenomena which notion that encapsulates the fixed-point structure finite group action on set. By applying this idea module, we provide results regarding enumeration matrices are invariant under...

Abstract In 2010, Rhoades proved that promotion on rectangular standard Young tableaux, together with the associated fake-degree polynomial, provides an instance of cyclic sieving phenomenon. We extend this result to m -tuples skew tableaux same shape, for fixed , subject condition th power polynomial evaluates nonnegative integers at roots unity. However, we are unable specify explicit group a...

Many index calculus algorithms generate multiplicative relations between smoothness basis elements by using a process called Sieving. This process allows to filter potential candidate relations very quickly, without spending too much time to consider bad candidates. However, from an asymptotic point of view, there is not much difference between sieving and straightforward testing of candidates....

The two classical hard problems underlying the security of lattice-based cryptography are the shortest vector problem (SVP) and the closest vector problem (CVP). For SVP, lattice sieving currently has the best (heuristic) asymptotic time complexity: in high dimensions d, sieving can solve SVP in time 2, using 2 memory [Becker– Ducas–Gama–Laarhoven, SODA’16]. The best heuristic time complexity t...

Filtration of molecules by nanometer-sized structures is ubiquitous in our everyday life, but our understanding of such molecular filtration processes is far less than desired. Until recently, one of the main reasons was the lack of experimental methods that can help provide detailed, microscopic pictures of molecule-nanostructure interactions. Several innovations in experimental methods, such ...

Many cryptographic protocols derive their security from the apparent computational intractability of the integer factorization problem. Currently, the best known integer-factoring algorithms run in subexponential time. Efficient parallel implementations of these algorithms constitute an important area of practical research. Most reported implementations use multi-core and/or distributed paralle...

The general number field sieve (GNFS) is the asymptotically fastest algorithm for factoring large integers. Its runtime depends on a good choice of a polynomial pair. In this article we present an improvement of the polynomial selection method of Montgomery and Murphy which has been used in recent GNFS records. 1. The polynomial selection method of Montgomery and Murphy In this section we brief...

Reiner, Stanton, and White [10] proved results regarding the enumeration of polygon dissections up to rotational symmetry. Eu and Fu [2] generalized these results to Cartan-Killing types other than A by means of actions of deformed Coxeter elements on cluster complexes of Fomin and Zelevinsky [6]. The ReinerStanton-White and Eu-Fu results were proven using direct counting arguments. We give rep...