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THE EXPECTED JAGGEDNESS OF ORDER IDEALS

Published online by Cambridge University Press:  15 March 2017

MELODY CHAN
Affiliation:
Department of Mathematics, Brown University, Providence, RI, USA; mtchan@math.brown.edu
SHAHRZAD HADDADAN
Affiliation:
Dipartimento di Informatica, Sapienza University of Rome, Rome, Italy; Shahrzad.Haddadan@gmail.com
SAM HOPKINS
Affiliation:
Department of Mathematics, MIT, Cambridge, MA, USA; shopkins@mit.edu
LUCA MOCI
Affiliation:
IMJ-PRG, Université Paris 7, Paris, France; lucamoci@hotmail.com

Abstract

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The jaggedness of an order ideal $I$ in a poset $P$ is the number of maximal elements in $I$ plus the number of minimal elements of $P$ not in $I$. A probability distribution on the set of order ideals of $P$ is toggle-symmetric if for every $p\in P$, the probability that $p$ is maximal in $I$ equals the probability that $p$ is minimal not in $I$. In this paper, we prove a formula for the expected jaggedness of an order ideal of $P$ under any toggle-symmetric probability distribution when $P$ is the poset of boxes in a skew Young diagram. Our result extends the main combinatorial theorem of Chan–López–Pflueger–Teixidor [Trans. Amer. Math. Soc., forthcoming. 2015, arXiv:1506.00516], who used an expected jaggedness computation as a key ingredient to prove an algebro-geometric formula; and it has applications to homomesies, in the sense of Propp–Roby, of the antichain cardinality statistic for order ideals in partially ordered sets.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2017

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