Publications (34)
Orientations, semiorders, arrangements, and parking functions
Sam Hopkins, David Perkinson
It is known that the Pak-Stanley labeling of the Shi hyperplane arrangement provides a bijection between the regions of the arrangement and parking functions. For any graph G, we d…
Upho lattices I: examples and non-examples of cores
Sam Hopkins
A poset is called upper homogeneous, or "upho," if every principal order filter of the poset is isomorphic to the whole poset. We study (finite type -graded) upho latti…
Another proof of Wilmes' conjecture
Sam Hopkins
We present a new proof of the monomial case of Wilmes' conjecture, which gives a formula for the coarsely-graded Betti numbers of the G-parking function ideal in terms of maximal p…
Cyclic Sieving for Staircase Plane Partitions via Crystals and Electrical Networks
Sam Hopkins, Jesse Kim, Stephan Pfannerer
The paper proves a cyclic sieving phenomenon for the promotion action on height‑two staircase plane partitions, using spin crystal representations and the bush basis of the electri…
Minuscule doppelgängers, the coincidental down-degree expectations property, and rowmotion
Sam Hopkins
We relate Reiner, Tenner, and Yong's coincidental down-degree expectations (CDE) property of posets to the minuscule doppelgänger pairs studied by Hamaker, Patrias, Pechenik, and…
Cyclic Sieving for Plane Partitions and Symmetry
Sam Hopkins
The cyclic sieving phenomenon of Reiner, Stanton, and White says that we can often count the fixed points of elements of a cyclic group acting on a combinatorial set by plugging ro…
Promotion of Kreweras words
Sam Hopkins, Martin Rubey
Kreweras words are words consisting of n A's, n B's, and n C's in which every prefix has at least as many A's as B's and at least as many A's as C's. Equivalently, a Kreweras word…
Root system chip-firing II: Central-firing
Pavel Galashin, Sam Hopkins, Thomas McConville +1
Jim Propp recently proposed a labeled version of chip-firing on a line and conjectured that this process is confluent from some initial configurations. This was proved by Hopkins-M…
Sorting via chip-firing
Sam Hopkins, Thomas McConville, James Propp
We investigate a variant of the chip-firing process on the infinite path graph: rather than treating the chips as indistinguishable, we label them with positive integers. To fire a…
Root system chip-firing I: Interval-firing
Pavel Galashin, Sam Hopkins, Thomas McConville +1
Jim Propp recently introduced a variant of chip-firing on a line where the chips are given distinct integer labels. Hopkins, McConville, and Propp showed that this process is confl…
Upho lattices II: ways of realizing a core
Sam Hopkins, Joel B. Lewis
A poset is called upper homogeneous, or "upho," if all of its principal order filters are isomorphic to the whole poset. In previous work of the first author, it was shown that eac…
Interlacing networks: birational RSK, the octahedron recurrence, and Schur function identities
Miriam Farber, Sam Hopkins, Wuttisak Trongsiriwat
Motivated by the problem of giving a bijective proof of the fact that the birational RSK correspondence satisfies the octahedron recurrence, we define interlacing networks, which a…
RSK via local transformations
Sam Hopkins
We explain how to define the Robinson-Schensted-Knuth (RSK) correspondence in terms of local transformations called "toggles." (This note, which is not intended for publication and…
A positive formula for the Ehrhart-like polynomials from root system chip-firing
Sam Hopkins, Alexander Postnikov
In earlier work in collaboration with Pavel Galashin and Thomas McConville we introduced a version of chip-firing for root systems. Our investigation of root system chip-firing led…
Fourientations and the Tutte Polynomial
Spencer Backman, Sam Hopkins
A fourientation of a graph is a choice for each edge of the graph whether to orient that edge in either direction, leave it unoriented, or biorient it. Fixing a total order on the…
On the -Enumeration of Barely Set-Valued Tableaux and Plane Partitions
Sam Hopkins, Alexander Lazar, Svante Linusson
Barely set-valued tableaux are a variant of Young tableaux in which one box contains two numbers as its entry. It has recently been discovered that there are product formulas enume…
Two -analogues of the tree inversion enumerator
Sam Hopkins
In this note, we introduce two -analogues and of the tree inversion enumerator . Although similar, and $\widetilde{I}_n(q,t)…
Order polynomial product formulas and poset dynamics
Sam Hopkins
We survey all known examples of finite posets whose order polynomials have product formulas, and we propose the heuristic that these are the same posets with good dynamical behavio…
The CDE property for minuscule lattices
Sam Hopkins
Reiner, Tenner, and Yong recently introduced the coincidental down-degree expectations (CDE) property for finite posets and showed that many nice posets are CDE. In this paper we f…
Restricted Birkhoff polytopes and Ehrhart period collapse
Per Alexandersson, Sam Hopkins, Gjergji Zaimi
We show that the polytopes obtained from the Birkhoff polytope by imposing additional inequalities restricting the "longest increasing subsequence" have Ehrhart quasi-polynomials w…
A note on Möbius functions of upho posets
Sam Hopkins
A poset is called upper homogeneous (or "upho") if every principal order filter of the poset is isomorphic to the whole poset. We observe that the rank and characteristic generatin…
Homomesy via Toggleability Statistics
Colin Defant, Sam Hopkins, Svetlana PoznanoviÄ +1
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…
Parking functions and tree inversions revisited
Petar Gaydarov, Sam Hopkins
Kreweras proved that the reversed sum enumerator for parking functions of length is equal to the inversion enumerator for labeled trees on vertices. Recently, Perkinson,…
Quantum integer-valued polynomials
Nate Harman, Sam Hopkins
We define a -deformation of the classical ring of integer-valued polynomials which we call the ring of quantum integer-valued polynomials. We show that this ring has a remarkabl…
The CDE property for skew vexillary permutations
Sam Hopkins
We prove a conjecture of Reiner, Tenner, and Yong which says that the initial weak order intervals corresponding to certain vexillary permutations have the coincidental down-degree…
Plane partitions of shifted double staircase shape
Sam Hopkins, Tri Lai
We give a product formula for the number of shifted plane partitions of shifted double staircase shape with bounded entries. This is the first new example of a family of shapes wit…
Pattern Avoidance in Poset Permutations
Sam Hopkins, Morgan Weiler
We extend the concept of pattern avoidance in permutations on a totally ordered set to pattern avoidance in permutations on partially ordered sets. The number of permutations on $P…
Fourientation activities and the Tutte polynomial
Spencer Backman, Sam Hopkins, Lorenzo Traldi
A fourientation of a graph is a choice for each edge of the graph whether to orient that edge in either direction, leave it unoriented, or biorient it. We may naturally view fo…
A note on statistical averages for oscillating tableaux
Sam Hopkins, Ingrid Zhang
We define a statistic called the weight of oscillating tableaux. Oscillating tableaux, a generalization of standard Young tableaux, are certain walks in Young's lattice of partitio…
Bigraphical Arrangements
Sam Hopkins, David Perkinson
We define the bigraphical arrangement of a graph and show that the Pak-Stanley labels of its regions are the parking functions of a closely related graph, thus proving conjectures…
The birational Lalanne-Kreweras involution
Sam Hopkins, Michael Joseph
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…
Combinatorial reciprocity for non-intersecting paths
Sam Hopkins, Gjergji Zaimi
We prove a combinatorial reciprocity theorem for the enumeration of non-intersecting paths in a linearly growing sequence of acyclic planar networks. We explain two applications of…
Symmetry of Narayana numbers and rowvacuation of root posets
Colin Defant, Sam Hopkins
For a Weyl group of rank , the -Catalan number is the number of antichains of the poset of positive roots, and the -Narayana numbers refine the -Catalan number by k…
The expected jaggedness of order ideals
Melody Chan, Shahrzad Haddadan, Sam Hopkins +1
The jaggedness of an order ideal I in a poset P is the number of maximal elements in I plus the number of minimal elements of P not in I. A probability distribution on the set of o…