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In his article "On orbits of antichains of positive roots" (European Journal of Combinatorics 30 (2009) 586–594; sorry, it's an Elsevier journal, so there's no freely available online version for me to give a link to), Dmitri Panyushev discusses an interesting self-map on the set of antichains of a finite poset (also discussed earlier by D.G. Fon-der-Flaass in "Orbits of antichains in ranked posets", European J. Combin. 14 (1993) 17–22, and even earlier by A. Brouwer and A. Schrijver in "On the period of an operator, defined on antichains", Math. Centrum Report ZW 24 (1974)).

Given an antichain $A$, we define $X(A)$ as the set of minimal elements of the complement of the order ideal generated by $A$.

One of Panyushev's conjectures, Conjecture 2.1(iii), asserts that for a certain class of posets the average $(1/|O|) \sum_{A \in O} |A|$ is the same for all $X$-orbits $O$, and gives a value for this average. This conjecture was recently proved by D. Armstrong, C. Stump, and H. Thomas in their article "A uniform bijection between nonnesting and noncrossing partitions", http://arxiv.org/abs/1101.1277 .

Does anyone know of any motivation behind Panyushev's conjecture about $(1/|O|) \sum_{A \in O} |A|$? Why would one be interested in this average? It's possible Panyushev may just have noticed the pattern numerically, with no particular theoretical purposes in mind. But I can't help feeling that he saw this conjecture as fitting into a larger story.

As a related question, is anyone in MathOverflow in touch with Panyushev? I tried sending him an email at the address listed in his article (asking the above question) but received no reply.

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I would bet that this is an empirical observation, with no special motivation. One interesting point about this self-map is that it satisfies the cyclic sieving phenomenon, as defined by Reiner, Stanton, and White in 2004. –  F. C. Dec 21 '11 at 13:01
    
There was a talk in this years FPSAC (2014), about averages over orbits related to posets. –  Per Alexandersson Aug 8 at 9:38

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