Dihedral Sieving Phenomena

Cyclic sieving is a well-known phenomenon where certain interesting polynomials, especially $q$-analogues, have useful interpretations related to actions and representations of the cyclic group. We propose a definition of sieving for an arbitrary group $G$ and study it for the dihedral group $I_2(n)$ of order $2n$. This requires understanding the generators of the representation ring of the dihedral group. For $n$ odd, we exhibit several instances of dihedral sieving which involve the generalized Fibonomial coefficients, recently studied by Amdeberhan, Chen, Moll, and Sagan. We also exhibit an instance of dihedral sieving involving Garsia and Haiman's $(q,t)$-Catalan numbers.Comment: 10 page

Effect of sieving on ex-situ soil respiration of soils from three land use types

This study aims to investigate the effect of sieving on ex situ soil respiration (CO2 flux) measurements from different land use types. We collected soils (0–10 cm) from arable, grassland and woodland sites, allocated them to either sieved (4-mm mesh, freshly sieved) or intact core treatments and incubated them in gas-tight jars for 40 days at 10 °C. Headspace gas was collected on days 1, 3, 17, 24, 31 and 38 and CO2 analysed. Our results showed that sieving (4 mm) did not significantly influence soil respiration measurements, probably because micro aggregates (< 0.25 mm) remain intact after sieving. However, soils collected from grassland soil released more CO2 compared with those collected from woodland and arable soils, irrespective of sieving treatments. The higher CO2 from grassland soil compared with woodland and arable soils was attributed to the differences in the water holding capacity and the quantity and stoichiometry of the organic matter between the three soils. We conclude that soils sieved prior to ex situ respiration experiments provide realistic respiration measurements. This finding lends support to soil scientists planning a sampling strategy that better represents the inhomogeneity of field conditions by pooling, homogenising and sieving samples, without fear of obtaining unrepresentative CO2 flux measurements caused by the disruption of soil architecture

Integer Factorization with a Neuromorphic Sieve

The bound to factor large integers is dominated by the computational effort to discover numbers that are smooth, typically performed by sieving a polynomial sequence. On a von Neumann architecture, sieving has log-log amortized time complexity to check each value for smoothness. This work presents a neuromorphic sieve that achieves a constant time check for smoothness by exploiting two characteristic properties of neuromorphic architectures: constant time synaptic integration and massively parallel computation. The approach is validated by modifying msieve, one of the fastest publicly available integer factorization implementations, to use the IBM Neurosynaptic System (NS1e) as a coprocessor for the sieving stage.Comment: Fixed typos in equation for modular roots (Section II, par. 6; Section III, par. 2) and phase calculation (Section IV, par 2

Cyclic sieving and cluster multicomplexes

Reiner, Stanton, and White \cite{RSWCSP} proved results regarding the enumeration of polygon dissections up to rotational symmetry. Eu and Fu \cite{EuFu} 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 \cite{FZY}. The Reiner-Stanton-White and Eu-Fu results were proven using direct counting arguments. We give representation theoretic proofs of closely related results using the notion of noncrossing and semi-noncrossing tableaux due to Pylyavskyy \cite{PN} as well as some geometric realizations of finite type cluster algebras due to Fomin and Zelevinsky \cite{FZClusterII}.Comment: To appear in Adv. Appl. Mat

Invariant tensors and the cyclic sieving phenomenon

We construct a large class of examples of the cyclic sieving phenomenon by expoiting the representation theory of semi-simple Lie algebras. Let $M$ be a finite dimensional representation of a semi-simple Lie algebra and let $B$ be the associated Kashiwara crystal. For $r\ge 0$, the triple $(X,c,P)$ which exhibits the cyclic sieving phenomenon is constructed as follows: the set $X$ is the set of isolated vertices in the crystal $\otimes^rB$; the map $c\colon X\rightarrow X$ is a generalisation of promotion acting on standard tableaux of rectangular shape and the polynomial $P$ is the fake degree of the Frobenius character of a representation of $\mathfrak{S}_r$ related to the natural action of $\mathfrak{S}_r$ on the subspace of invariant tensors in $\otimes^rM$. Taking $M$ to be the defining representation of $\mathrm{SL}(n)$ gives the cyclic sieving phenomenon for rectangular tableaux