The Catalan case of Armstrong's conjecture on simultaneous core partitions

Richard P. Stanley; Fabrizio Zanello

The Catalan case of Armstrong's conjecture on simultaneous core partitions

Combinatorics 2015-04-03 v2 Commutative Algebra Number Theory

Authors: Richard P. Stanley , Fabrizio Zanello

Abstract

A beautiful recent conjecture of D. Armstrong predicts the average size of a partition that is simultaneously an $s$ -core and a $t$ -core, where $s$ and $t$ are coprime. Our goal is to prove this conjecture when $t=s+1$ . These simultaneous $(s,s+1)$ -core partitions, which are enumerated by Catalan numbers, have average size $\binom{s+1}{3}/2$ .

Keywords

simultaneous core partitions Armstrong's conjecture Catalan numbers coprime integers partition size combinatorics

Cite

@article{arxiv.1312.4352,
  title  = {The Catalan case of Armstrong's conjecture on simultaneous core partitions},
  author = {Richard P. Stanley and Fabrizio Zanello},
  journal= {arXiv preprint arXiv:1312.4352},
  year   = {2015}
}

Comments

Some changes in response to the referee's comments. To appear in the SIAM J. on Discrete Math

The Catalan case of Armstrong's conjecture on simultaneous core partitions

Abstract

Keywords

Cite

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