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THE EXPECTED JAGGEDNESS OF ORDER IDEALS

Part of: Ordered sets Computational aspects in algebraic geometry Algebraic combinatorics

Published online by Cambridge University Press:  15 March 2017

MELODY CHAN ,
SHAHRZAD HADDADAN ,
SAM HOPKINS  and
LUCA MOCI
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.

MSC classification

Primary: 05E18: Group actions on combinatorial structures 06A07: Combinatorics of partially ordered sets
Secondary: 14Q05: Curves
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 5 , 2017 , e9
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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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