Strange expectations and the Winnie-the-Pooh problem

@article{Thiel2020StrangeEA,
  title={Strange expectations and the Winnie-the-Pooh problem},
  author={Mark G. Thiel and Nathan Williams},
  journal={J. Comb. Theory, Ser. A},
  year={2020},
  volume={176},
  pages={105298}
}
Motivated by the study of simultaneous cores, we give three proofs (in varying levels of generality) for the expected norm of a weight in a highest weight representation of a complex simple Lie algebra. First, we argue directly using the polynomial method and the Weyl character formula. Second, we use the combinatorics of semistandard tableaux to obtain the result in type A. Third, and most interestingly, we relate this problem to the "Winnie-the-Pooh problem" regarding orthogonal… Expand
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References

SHOWING 1-10 OF 31 REFERENCES
Cataland: Why the Fuß?
The three main objects in noncrossing Catalan combinatorics associated to a finite Coxeter system are noncrossing partitions, clusters, and sortable elements. The first two of these have knownExpand
Asymptotic normality of the major index on standard tableaux
We consider the distribution of the major index on standard tableaux of arbitrary straight shape and certain skew shapes. We use cumulants to classify all possible limit laws for any sequence of suchExpand
Conjectures on the Quotient Ring by Diagonal Invariants
AbstractWe formulate a series of conjectures (and a few theorems) on the quotient of the polynomial ring $$\mathbb{Q}[x_1 , \ldots ,x_n ,y_1 , \ldots ,y_n ]$$ in two sets of variables by the idealExpand
Results and conjectures on simultaneous core partitions
TLDR
It is proved that 2n- and (2mn+1)-core partitions correspond naturally to dominant alcoves in the m-Shi arrangement of type C"n, generalizing a result of Fishel-Vazirani for type A. Expand
Bruhat Lattices, Plane Partition Generating Functions, and Minuscule Representations
TLDR
The Bruhat posets (arising from Weyl groups) which are lattices are classified to show that certain combinatorially defined generating functions associated to these lattices satisfy certain identities. Expand
Mutually unbiased bases and orthogonal decompositions of Lie algebras
TLDR
Monomial MUBs are defined, a class of which all known MUB constructions are members, and it is shown that for n = 6 there are at most three monomial Mubs. Expand
Catalan numbers for complex reflection groups
We construct $(q,t)$-Catalan polynomials and $q$-Fuss-Catalan polynomials for any irreducible complex reflection group $W$. The two main ingredients in this construction are Rouquier's formulation ofExpand
The Catalan Case of Armstrong's Conjecture on Simultaneous Core Partitions
TLDR
This paper aims to prove the conjecture that the average size of a partition that is simultaneously an $s-core and a $t-core, where $s and $t$ are coprime, when $t=s+1$. Expand
Armstrong's conjecture for (k, mk+1)-core partitions
  • A. Aggarwal
  • Computer Science, Mathematics
  • Eur. J. Comb.
  • 2015
TLDR
This paper uses a variant of Stanley and Zanello's method to establish Armstrong's conjecture in the more general setting where a divides b - 1. Expand
Lattice Points and Simultaneous Core Partitions
  • P. Johnson
  • Computer Science, Mathematics
  • Electron. J. Comb.
  • 2018
TLDR
Using Ehrhart theory and Euler-Maclaurin theory, it is proved that Armstrong's conjecture that the average size of an $(a,b)$-core is $(a+b+1)(a-1)(b-1)/24$. Expand
...
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