# Results and conjectures on simultaneous core partitions

@article{Armstrong2014ResultsAC,
title={Results and conjectures on simultaneous core partitions},
author={Drew Armstrong and Christopher R. H. Hanusa and Brant C. Jones},
journal={Eur. J. Comb.},
year={2014},
volume={41},
pages={205-220}
}
• Published 2 August 2013
• Mathematics
• Eur. J. Comb.

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## References

SHOWING 1-10 OF 33 REFERENCES

### A bijection between (bounded) dominant Shi regions and core partitions

• Mathematics
• 2010
It is well-known that Catalan numbers $C_n = \frac{1}{ n+1} \binom{2n}{n}$ count the number of dominant regions in the Shi arrangement of type $A$, and that they also count partitions which are both

### On a refinement of the generalized Catalan numbers for Weyl groups

Let Φ be an irreducible crystallographic root system with Weyl group W, coroot lattice Q and Coxeter number h, spanning a Euclidean space V, and let m be a positive integer. It is known that the set

### Generalized Catalan Numbers, Weyl Groups and Arrangements of Hyperplanes

For an irreducible, crystallographic root system Φ in a Euclidean space V and a positive integer m, the arrangement of hyperplanes in V given by the affine equations (α, x) = k, for α ∈ Φ and k = 0,

### Rational Associahedra and Noncrossing Partitions

• Mathematics
Electron. J. Comb.
• 2013
It is proved that Ass (a,b) is shellable and nice product formulas for its h-vector and f-vector are given and a rational generalization of noncrossing perfect matchings of [2n] is defined.

### On Two Presentations of the Affine Weyl Groups of Classical Types

• Mathematics
• 1999
Abstract The main result of the paper is to get the transition formulae between the alcove form and the permutation form of w ∈ Wa, where Wa is an affine Weyl group of classical type. On the other

### Surveys in Combinatorics 2011: The cyclic sieving phenomenon: a survey

The cyclic sieving phenomenon was defined by Reiner, Stanton, and White in a 2004 paper. Let X be a finite set, C be a finite cyclic group acting on X, and f(q) be a polynomial in q with nonnegative

### Combinatory Analysis

WHEN the first volume of this work was noticed in these columns, the reviewer of that volume expressed the hope that the second would not be long delayed. This hope has been fulfilled, and the reader