The jaggedness of an order ideal in a poset is the number of maximal elements in plus the number of minimal elements of not in . A probability distribution on the set of order ideals of is toggle-symmetric if for every , the probability that is maximal in equals the probability that is minimal not in . In this paper, we prove a formula for the expected jaggedness of an order ideal of under any toggle-symmetric probability distribution when 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.

Chan, M., Haddadan, S., Hopkins, S., Moci, L. (2017). The expected jaggedness of order ideals. FORUM OF MATHEMATICS. SIGMA, 5, 1-27 [10.1017/fms.2017.5].

### The expected jaggedness of order ideals

#### Abstract

The jaggedness of an order ideal in a poset is the number of maximal elements in plus the number of minimal elements of not in . A probability distribution on the set of order ideals of is toggle-symmetric if for every , the probability that is maximal in equals the probability that is minimal not in . In this paper, we prove a formula for the expected jaggedness of an order ideal of under any toggle-symmetric probability distribution when 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.
##### Scheda breve Scheda completa Scheda completa (DC)
2017
Chan, M., Haddadan, S., Hopkins, S., Moci, L. (2017). The expected jaggedness of order ideals. FORUM OF MATHEMATICS. SIGMA, 5, 1-27 [10.1017/fms.2017.5].
File in questo prodotto:
File
expected_jaggedness_of_order_ideals.pdf

accesso aperto

Tipo: Versione (PDF) editoriale
Licenza: Licenza per Accesso Aperto. Creative Commons Attribuzione (CCBY)
Dimensione 332.75 kB
Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11585/676930`