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Register a new userFoatic actions of the symmetric group and fixed-point homomesy
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We study maps on the set of permutations of n generated by the Renyi-Foata map intertwined with other dihedral symmetries (of a permutation considered as a 0-1 matrix). Iterating these maps leads to dynamical systems that in some cases exhibit interesting orbit structures, e.g., every orbit size being a power of two, and homomesic statistics (ones which have the same average over each orbit). In particular, the number of fixed points (aka 1-cycles) of a permutation appears to be homomesic with respect to three of these maps, even in one case where the orbit structures are far from nice. For the most interesting such Foatic action, we give a heap analysis and recursive structure that allows us to prove the fixed-point homomesy and orbit properties, but two other cases remain conjectural.
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Many invertible actions $tau$ on a set ${mathcal{S}}$ of combinatorial objects, along with a natural statistic $f$ on ${mathcal{S}}$, exhibit the following property which we dub textbf{homomesy}: the average of $f$ over each $tau$-orbit in ${mathcal{S}}$ is the same as the average of $f$ over the whole set ${mathcal{S}}$. This phenomenon was first noticed by Panyushev in 2007 in the context of the rowmotion action on the set of antichains of a root poset; Armstrong, Stump, and Thomas proved Panyushevs conjecture in 2011. We describe a theoretical framework for results of this kind that applies more broadly, giving examples in a variety of contexts. These include linear actions on vector spaces, sandpile dynamics, Suters action on certain subposets of Youngs Lattice, Lyness 5-cycles, promotion of rectangular semi-standard Young tableaux, and the rowmotion and promotion actions on certain posets. We give a detailed description of the latter situation for products of two chains.
This paper presents a generalization of the sandpile model, called the parallel symmetric sandpile model, which inherits the rules of the symmetric sandpile model and implements them in parallel. In this new model, at each step the collapsing of the collapsible columns happens at the same time and one collapsible column is able to collapse on the left or on the right but not both. We prove that the set of forms of fixed points of the symmetric sandpile model is the same as the one of that model using parallel update scheme by constructing explicitly the way (in the parallel update scheme) to reach the form of an arbitrary fixed point of the sequential model.
Given a group $Gamma$ acting on a set $X$, a $k$-coloring $phi:Xto{1,dots,k}$ of $X$ is distinguishing with respect to $Gamma$ if the only $gammain Gamma$ that fixes $phi$ is the identity action. The distinguishing number of the action $Gamma$, denoted $D_{Gamma}(X)$, is then the smallest positive integer $k$ such that there is a distinguishing $k$-coloring of $X$ with respect to $Gamma$. This notion has been studied in a number of settings, but by far the largest body of work has been concerned with finding the distinguishing number of the action of the automorphism group of a graph $G$ upon its vertex set, which is referred to as the distinguishing number of $G$. The distinguishing number of a group action is a measure of how difficult it is to break all of the permutations arising from that action. In this paper, we aim to further differentiate the resilience of group actions with the same distinguishing number. In particular, we introduce a precoloring extension framework to address this issue. A set $S subseteq X$ is a fixing set for $Gamma$ if for every non-identity element $gamma in Gamma$ there is an element $s in S$ such that $gamma(s) eq s$. The distinguishing extension number $operatorname{ext}_D(X,Gamma;k)$ is the minimum number $m$ such that for all fixing sets $W subseteq X$ with $|W| geq m$, every $k$-coloring $c : X setminus W to [k]$ can be extended to a $k$-coloring that distinguishes $X$. In this paper, we prove that $operatorname{ext}_D(mathbb{R},operatorname{Aut}(mathbb{R}),2) =4$, where $operatorname{Aut}(mathbb{R})$ is comprised of compositions of translations and reflections. We also consider the distinguishing extension number of the circle and (finite) cycles, obtaining several exact results and bounds.
We provide combinatorial interpretation for the $gamma$-coefficients of the basic Eulerian polynomials that enumerate permutations by the excedance statistic and the major index as well as the corresponding $gamma$-coefficients for derangements. Our results refine the classical $gamma$-positivity results for the Eulerian polynomials and the derangement polynomials. The main tools are Brandens modified Foata--Strehl action on permutations and the recent triple statistic (des, rix,aid) equidistibuted with (exc, fix, maj).
We prove that the expected number of braid moves in the commutation class of the reduced word $(s_1 s_2 cdots s_{n-1})(s_1 s_2 cdots s_{n-2}) cdots (s_1 s_2)(s_1)$ for the long element in the symmetric group $mathfrak{S}_n$ is one. This is a variant of a similar result by V. Reiner, who proved that the expected number of braid moves in a random reduced word for the long element is one. The proof is bijective and uses X. Viennots theory of heaps and variants of the promotion operator. In addition, we provide a refinement of this result on orbits under the action of even and odd promotion operators. This gives an example of a homomesy for a nonabelian (dihedral) group that is not induced by an abelian subgroup. Our techniques extend to more general posets and to other statistics.
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