Motivation behind Panyushev's "constant-averages-along-orbits" conjecture In his article "On orbits of antichains of positive roots" (European Journal of Combinatorics 30 (2009) 586–594, 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.
 A: In "The root poset and its relatives" (https://arxiv.org/abs/math/0502385) Panyushev established (see Corollary 3.4) that the average size of an antichain of the root poset $\Phi^+$ of an irreducible crystallographic root system $\Phi$ is $n/2$, where $n$ is the number of simple roots of $\Phi$. He did this just by observing that the "$\Phi$-Narayana polynomial", i.e., the generating function for antichains of $\Phi^+$ by cardinality, is palindromic, something he in fact observed in his earlier paper "ad-nilpotent ideals of a Borel subalgebra: generators and duality" (https://www.sciencedirect.com/science/article/pii/S0021869303006380) (see Section 6).
In the "ad-nilpotent ideals" paper he is very interested in the fact that the Narayana polynomials are palindromic and uses this palindromicness as evidence in favor of a certain conjectured "duality" for the ad-nilpotent ideals (which correspond bijectively to antichains of the root post). In particular he conjectures (see Conjecture 6.1) the existence of a natural involution on the set of antichains which would send an antichain of cardinality $k$ to an antichain of cardinality $n-k$. As far as I know, this conjecture remains open (see my MO question Panyushev's conjectured duality for root poset antichains).
He couldn't prove the existence of that duality, but in the "On orbits of antichains of positive roots" paper (https://arxiv.org/abs/0711.3353) he accomplished something close using "rowmotion"/"the Fon-der-Flaass operator". Namely, he conjectured a specific way to partition the set of antichains of $\Phi^+$ into "small" sets (of size dividing $2h$ where $h$ is the Coxeter number of $\Phi$) such that in each such set the average size is $n/2$. (So the conjectured involution on the set of antichains would be doing the same thing except with sets of size dividing $2$ instead of $2h$.)
In this way I view Panyushev's homomesy conjecture as an extension of his investigation of the "duality" property for ad-nilpotent ideals, which evolved over the course of the three papers referenced above.
EDIT: Here is some further "context"/speculation:
Let $\Phi$ an irreducible crystallographic root system with Weyl group $W$. As mentioned above, the $\Phi$-Narayana number $N_k(\Phi)$ is the number of antichains of the root poset $\Phi^+$ of cardinality $k$. And the $\Phi$-Narayana polynomial is then $N(\Phi;q) = \sum_{k=0}^{n} N_k(\Phi)q^k$. Note that $N(\Phi;1)=\mathrm{Cat}(\Phi)$ is the $\Phi$-Catalan number.
The previous paragraph gave a "nonnesting" description of $N(\Phi;q)$. There is also a "noncrossing" description. Namely, recall that the lattice of noncrossing partitions of $\Phi$, denoted $NC(\Phi)$, is the induced subposet of the absolute order on $W$ below $c$, where $c$ is any fixed Coxeter element of $W$. Then $NC(\Phi)$ is a graded poset, and the Narayana number $N_k(\Phi)$ is also the number of elements of $NC(\Phi)$ at rank $k$ (so $N(\Phi;q)$ is the rank generating function for $NC(\Phi)$). For these (and other) descriptions of $N(\Phi;q)$, see Theorem 5.9 of Fomin-Reading "Root systems and generalized associahedra" (https://arxiv.org/abs/math/0505518).
Now, as mentioned, Panyushev was very interested in the fact that $N(\Phi;q)$ is palindromic, which is not at all obvious from the nonnesting description. In particular he was looking for a bijective (in fact, involutive) proof of this fact. But there is a nice way to see that $N(\Phi;q)$ is palindromic from the noncrossing description. Namely, there is the Kreweras complementation map $\mathrm{Krew}\colon w \mapsto cw^{-1}$ on $NC(\Phi)$, which takes an element of rank $k$ to an element of rank $n-k$.
Note that while $\mathrm{Krew}$ does provide a bijective proof that $N(\Phi;q)$ is palindromic, $\mathrm{Krew}$ is not an involution (it has order $h$ or $2h$). In fact, here's where the connection to Panyushev's "rowmotion" action comes in: elements of $NC(\Phi)$ under $\mathrm{Krew}$ are in equivariant bijection with antichains of $\Phi^+$ under rowmotion. This was conjectured by Bessis-Reiner (https://arxiv.org/abs/math/0701792) and proved by Armstrong-Stump-Thomas (https://arxiv.org/abs/1101.1277).
As far as I can tell, Panyushev did not know at the time that his action was the "same" as Kreweras complementation. But I do think it's interesting that he came up with his action with the goal of understanding why $N(\Phi;q)$ is palindromic on the nonnesting side (or at least why the average antichain cardinality is $n/2$), while the Kreweras complementation proves this on the noncrossing side.
