## 86 Citations

### Advances in the Theory of Cores and Simultaneous Core Partitions

- MathematicsAm. Math. Mon.
- 2017

A review of five recent papers in this area by undergraduates, ranging from generating functions and modular forms to more combinatorial tools such as abaci, posets, and lattice paths, give a flavor of the richness of the subject.

### Lattice Points and Simultaneous Core Partitions

- MathematicsElectron. J. Comb.
- 2018

Using Ehrhart theory and Euler-Maclaurin theory, it is proved that Armstrong's conjecture that the average size of an $(a,b)$-core is $(a+b+1)(a-1)(b-1)/24$.

### Core partitions into distinct parts and an analog of Euler's theorem

- MathematicsEur. J. Comb.
- 2016

### Johnson's bijections and their application to counting simultaneous core partitions

- MathematicsEur. J. Comb.
- 2019

### The Corners of Core Partitions

- MathematicsSIAM J. Discret. Math.
- 2018

It is proved that the antistitches of a rational Dyck path are in bijection with the segments of structure sets of the corresponding core partition; therefore the corners of a core partition can be counted by the number of stitches or antistitch.

### RANK COMPLEMENT OF RATIONAL DYCK PATHS AND CONJUGATION OF (m;n)-CORE PARTITIONS

- Mathematics
- 2015

Given a coprime pair (m;n) of positive integers, rational Catalan numbers 1 m+n m+n m;n counts two combinatorial objects: rational (m;n)-Dyck paths are lattice paths in the m n rectangle that never…

### Explicit Expressions for the Variance and Higher Moments of the Size of a Simultaneous Core Partition and its Limiting Distribution

- Mathematics
- 2015

Jaclyn Anderson proved that if s and t are relatively prime positive integers, then there are exactly (s+t-1)!/(s!t!) partitions whose set of hook-lengths is disjoint from the set {s,t}. Drew…

## References

SHOWING 1-10 OF 33 REFERENCES

### A bijection between dominant Shi regions and core partitions

- MathematicsEur. J. Comb.
- 2010

### 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

- Mathematics
- 2004

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

- Mathematics
- 2004

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

- MathematicsElectron. 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

- Mathematics
- 2010

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

- HistoryNature
- 1917

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…