Nathan Williams

Promotion and rowmotion

We present an equivariant bijection between two actions-promotion and rowmotion-on order ideals in certain posets. This bijection simultaneously generalizes a result of R. Stanley concerning promotion on the linear extensions of two disjoint chains and certain cases of recent work of D. Armstrong, C. Stump, and H. Thomas on noncrossing and nonnesting partitions. We apply this bijection to several classes of posets, obtaining equivariant bijections to various known objects under rotation. We extend the same idea to give an equivariant bijection between alternating sign matrices under rowmotion and under B. Wielands gyration. Finally, we define two actions with related orders on alternating sign matrices and totally symmetric self-complementary plane partitions.

Sweeping up zeta

We repurpose the main theorem of Thomas and Williams (J Algebr Comb 39(2):225–246, 2014) to prove that modular sweep maps are bijective. We construct the inverse of the modular sweep map by passing through an intermediary set of equitable partitions; motivated by an analogy to stable marriages, we prove that the set of equitable partitions for a fixed word forms a distributive lattice when ordered component wise. We conclude that the general sweep maps defined in Armstrong et al. (Adv Math 284:159–185, 2015) are bijective. As a special case of particular interest, this gives the first proof that the zeta map on rational Dyck paths is a bijection.

Stable multivariate W-Eulerian polynomials

Abstract We prove a multivariate strengthening of Brentiʼs result that every root of the Eulerian polynomial of type B is real. Our proof combines a refinement of the descent statistic for signed permutations with the notion of real stability—a generalization of real-rootedness to polynomials in multiple variables. The key is that our refined multivariate Eulerian polynomials satisfy a recurrence given by a stability-preserving linear operator. Our results extend naturally to colored permutations, and we also give stable generalizations of recent real-rootedness results due to Dilks, Petersen, and Stembridge on affine Eulerian polynomials of types A and C . Finally, although we are not able to settle Brentiʼs real-rootedness conjecture for Eulerian polynomials of type D , nor prove a companion conjecture of Dilks, Petersen, and Stembridge for affine Eulerian polynomials of types B and D , we indicate some methods of attack and pose some related open problems.

Noncrossing partitions and Bruhat order

We prove that the restriction of Bruhat order to noncrossing partitions in type

Braid moves in commutation classes of the symmetric group

A_n

Rational Associahedra and Noncrossing Partitions

for the Coxeter element

Cyclic symmetry of the scaled simplex

c=s_1s_2 ...s_n

Doppelg\"angers: Bijections of Plane Partitions

forms a distributive lattice isomorphic to the order ideals of the root poset ordered by inclusion. Motivated by the change-of-basis from the graphical basis of the Temperley-Lieb algebra to the image of the simple elements of the dual braid monoid, we extend this bijection to other Coxeter elements using certain canonical factorizations. In particular, we give new bijections---fixing the set of reflections---between noncrossing partitions associated to distinct Coxeter elements.

Automata, reduced words, and Garside shadows in Coxeter groups

We prove that the expected number of braid moves in the commutation class of the reduced word (s1s2sn1)(s1s2sn2)(s1s2)(s1) for the long element in the symmetric group Sn 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.