# Order polynomial product formulas and poset dynamics

@article{Hopkins2020OrderPP, title={Order polynomial product formulas and poset dynamics}, author={Sam Hopkins}, journal={arXiv: Combinatorics}, year={2020} }

We survey all known examples of finite posets whose order polynomials have product formulas, and we put forward a heuristic which says that these are the same posets which have good dynamical behavior. Here the dynamics in question are the actions of promotion on the linear extensions of the poset, and rowmotion on the P-partitions of the poset.

## 10 Citations

Dynamical Algebraic Combinatorics (Online)

- 2021

Rowmotion and Homomesy. Rowmotion was introduced by Duchet in [Duc74]; studied for the Boolean lattice (and the product of two chains) by Brouwer and Schrijver [BS74, Bro75]; and (still for the…

Promotion and cyclic sieving on families of SSYT

- MathematicsArkiv för Matematik
- 2021

We examine a few families of semistandard Young tableaux, for which we observe the cyclic sieving phenomenon under promotion.
The first family we consider consists of stretched hook shapes, where we…

Symmetry of Narayana Numbers and Rowvacuation of Root Posets

- MathematicsForum of Mathematics, Sigma
- 2021

Abstract For a Weyl group W of rank r, the W-Catalan number is the number of antichains of the poset of positive roots, and the W-Narayana numbers refine the W-Catalan number by keeping track of the…

Promotion of Kreweras words

- MathematicsSelecta Mathematica
- 2021

Kreweras words are words consisting of n$$\mathrm {A}$$
A
’s, n$$\mathrm {B}$$
B
’s, and n$$\mathrm {C}$$
C
’s in which every prefix has at least as many $$\mathrm {A}$$
A
’s as $$\mathrm…

On the interaction of the Coxeter transformation and the rowmotion bijection

- Mathematics
- 2022

Let P be a finite poset and L the associated distributive lattice of order ideals of P . Let ρ denote the rowmotion bijection of the order ideals of P viewed as a permutation matrix and C the Coxeter…

A recursive approach for the enumeration of the homomorphisms from a poset $P$ to the chain $C_3$

- Mathematics
- 2021

Let H(P,C3) be the set of order homomorphisms from a poset P to the chain C3 = 1 < 2 < 3. We develop a recursive approach for the calculation of the cardinality of H(P,C3), and we apply it on several…

FFLV polytopes for odd symplectic Lie algebras

- Mathematics
- 2021

We consider “odd symplectic Lie algebras” defined in terms of maximal rank skew-symmetric forms. We provide FFLV polytopes for these algebras and prove their standard properties. In particular, we…

Homomesy via Toggleability Statistics

- Mathematics
- 2021

The rowmotion operator acting on the set of order ideals of a finite poset has been the focus of a significant amount of recent research. One of the major goals has been to exhibit homomesies:…

Plane partitions of shifted double staircase shape

- Mathematics, Computer ScienceJ. Comb. Theory, Ser. A
- 2021

This is the first new example of a family of shapes with a plane partition product formula in many years, based on the theory of lozenge tilings, that gives a product formula for the number of shifted plane partitions of shifted double staircase shape with bounded entries.

The birational Lalanne-Kreweras involution

- Mathematics
- 2020

The Lalanne–Kreweras involution is an involution on the set of Dyck paths which combinatorially exhibits the symmetry of the number of valleys and major index statistics. We define piecewise-linear…

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