• 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. 
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Methods Citations
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References

SHOWING 1-10 OF 98 REFERENCES

Orbits of Antichains in Ranked Posets

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.

Birational Rowmotion and Coxeter-Motion on Minuscule Posets

  • S. Okada
  • Mathematics
    Electron. J. Comb.
  • 2021
As a generalization of birational promotion on a product of two chains, birational Coxeter-motion is introduced, and it is proved that it enjoys periodicity and file homomesy.

Orbits of antichains revisited

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

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

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

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…
...