• Corpus ID: 219179711

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}
}
  • S. Hopkins
  • Published 2 June 2020
  • Mathematics
  • arXiv: Combinatorics
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. 

Figures from this paper

Dynamical Algebraic Combinatorics (Online)
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
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
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
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
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 $C_3$
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
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
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
TLDR
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
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

References

SHOWING 1-10 OF 98 REFERENCES
Orbits of Antichains in Ranked Posets
TLDR
A duality relation on orbits of this permutation f of antichains of a ranked poset P is found, which is used for proving a conjecture by M. Deza and K. Fukuda about the set of lower units of any monotone boolean function on P to its upper zeros.
Orbits of antichains revisited
TLDR
A new treatment of the permutation f of antichains in ranked posets moving the set of lower units of a monotone Boolean function to the setof its upper zeros is presented.
Rowmotion Orbits of Trapezoid Posets
Rowmotion is an invertible operator on the order ideals of a poset which has been extensively studied and is well understood for the rectangle poset. In this paper, we show that rowmotion is
Hook length property of d-complete posets via q-integrals
TLDR
A new proof of the hook length formula usingq-integrals is given, which expresses the P-partition generating function for each case as a $q$-integral and then is evaluated using partial fraction expansion identities.
Counting linear extensions
We survey the problem of counting the number of linear extensions of a partially ordered set. We show that this problem is #P-complete, settling a long-standing open question. This result is
Iterative Properties of Birational Rowmotion II: Rectangles and Triangles
TLDR
The finite order of birational rowmotion on many posets can be considered an analogue to Zamolodchikov periodicity, and commented on suspected, but so far enigmatic, connections to the theory of root posets.
Promotion and rowmotion
TLDR
An equivariant bijection between two actions-promotion and rowmotion-on order ideals in certain posets is presented and two actions with related orders on alternating sign matrices and totally symmetric self-complementary plane partitions are defined.
Promotion and cyclic sieving via webs
We show that Schützenberger’s promotion on two and three row rectangular Young tableaux can be realized as cyclic rotation of certain planar graphs introduced by Kuperberg. Moreover, following work
Conjectures on the Quotient Ring by Diagonal Invariants
AbstractWe formulate a series of conjectures (and a few theorems) on the quotient of the polynomial ring $$\mathbb{Q}[x_1 , \ldots ,x_n ,y_1 , \ldots ,y_n ]$$ in two sets of variables by the ideal
Rowmotion and increasing labeling promotion
TLDR
A generalization of K-promotion to the setting of arbitrary increasing labelings of any finite poset with given restrictions on the labels shows it corresponds to a toggle group action the authors call toggle-prom promotion on order ideals of an associated poset when the restrictions on labels are particularly nice.
...
1
2
3
4
5
...