Papers by Nathan Williams
The Electronic Journal of Combinatorics
We generalize the Tamari lattice by extending the notions of $231$-avoiding permutations, noncros... more We generalize the Tamari lattice by extending the notions of $231$-avoiding permutations, noncrossing set partitions, and nonnesting set partitions to parabolic quotients of the symmetric group $\mathfrak{S}_{n}$. We show bijectively that these three objects are equinumerous. We show how to extend these constructions to parabolic quotients of any finite Coxeter group. The main ingredient is a certain aligned condition of inversion sets; a concept which can in fact be generalized to any reduced expression of any element in any (not necessarily finite) Coxeter group.
International Mathematics Research Notices
We say two posets are doppelgängers if they have the same number of P-partitions of each height k... more We say two posets are doppelgängers if they have the same number of P-partitions of each height k. We give a uniform framework for bijective proofs that posets are doppelgängers by synthesizing K-theoretic Schubert calculus techniques of H. Thomas and A. Yong with M. Haiman’s rectification bijection and an observation of R. Proctor. Geometrically, these bijections reflect the rational equivalence of certain subvarieties of minuscule flag manifolds. As a special case, we provide the 1st bijective proof of a 1983 theorem of R. Proctor—that plane partitions of height k in a rectangle are equinumerous with plane partitions of height k in a shifted trapezoid.
Selecta Mathematica
We repurpose the main theorem of [TW14] to prove that modular sweep maps are bijective. We constr... more We repurpose the main theorem of [TW14] 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 componentwise. We conclude that the general sweep maps defined in [ALW15] 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.
European Journal of Combinatorics
We prove that the expected number of braid moves in the commutation class of the reduced word (s ... more We prove that the expected number of braid moves in the commutation class of the reduced word (s 1 s 2 • • • s n−1)(s 1 s 2 • • • s n−2) • • • (s 1 s 2)(s 1) 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. Viennot's 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.
European Journal of Combinatorics, 2016
We prove that the restriction of Bruhat order to noncrossing partitions in type An for the Coxete... more We prove that the restriction of Bruhat order to noncrossing partitions in type An for the Coxeter element c = s1s2 • • • sn 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.
Journal of Algebraic Combinatorics
Let gcd(a, b) = 1. J. Olsson and D. Stanton proved that the maximum number of boxes in a simultan... more Let gcd(a, b) = 1. J. Olsson and D. Stanton proved that the maximum number of boxes in a simultaneous (a, b)-core is max λ∈core(a,b) (size(λ)) = (a 2 − 1)(b 2 − 1) 24 and that this maximum is achieved by a unique core. P. Johnson combined Ehrhart theory with the polynomial method to prove D. Armstrong's conjecture that the expected number of boxes in a simultaneous (a, b)-core is E λ∈core(a,b) (size(λ)) = (a − 1)(b − 1)(a + b + 1) 24. We extend Johnson's method to compute the variance to be V λ∈core(a,b) (size(λ)) = ab(a − 1)(b − 1)(a + b)(a + b + 1) 1440 , and also prove polynomiality of all moments. By extending the definitions of "simultaneous cores" and "number of boxes" to affine Weyl groups, we give uniform generalizations of all three formulae above to simply laced affine types. We further B Nathan Williams
We repurpose the main theorem of [TW14] to prove that modular sweep maps are bijective. We constr... more We repurpose the main theorem of [TW14] 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 componentwise. We conclude that the general sweep maps defined in [ALW14b] 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.
Let gcd(a, b) = 1. J. Olsson and D. Stanton proved that the maximum number of boxes in a simultan... more Let gcd(a, b) = 1. J. Olsson and D. Stanton proved that the maximum number of boxes in a simultaneous (a, b)-core is max λ∈core(a,b) (size(λ)) = (a 2 − 1)(b 2 − 1) 24 , and that this maximum was achieved by a unique core. P. Johnson combined Ehrhart theory with the polynomial method to prove D. Armstrong's conjecture that the expected number of boxes in a simultaneous (a, b)-core is E λ∈core(a,b) (size(λ)) = (a − 1)(b − 1)(a + b + 1) 24. We extend P. Johnson's method to compute the variance to be V λ∈core(a,b) (size(λ)) = ab(a − 1)(b − 1)(a + b)(a + b + 1) 1440 , and also prove polynomiality of all moments. By extending the definitions of " simultaneous cores " and " number of boxes " to affine Weyl groups, we give uniform generalizations of all three formulae above to simply-laced affine types. We further explain the appearance of the number 24 using the " strange formula " of H. Freudenthal and H. de Vries.
We use a projection argument to uniformly prove that $W$-permutahedra and $W$-associahedra have t... more We use a projection argument to uniformly prove that $W$-permutahedra and $W$-associahedra have the property that if $v,v'$ are two vertices on the same face $f$, then any geodesic between $v$ and $v'$ does not leave $f$. In type $A$, we show that our geometric projection recovers a slight modification of the combinatorial projection given by D. Sleator, R. Tarjan, and W. Thurston.
Journal of Algebraic Combinatorics, 2014
Let Z k m consist of the m k alcoves contained in the m-fold dilation of the fundamental alcove o... more Let Z k m consist of the m k alcoves contained in the m-fold dilation of the fundamental alcove of the type A k affine hyperplane arrangement. As the fundamental alcove has a cyclic symmetry of order (k + 1), so does Z k m . By bijectively exchanging the natural poset structure of Z k m for a natural cyclic action on a set of words, we prove that (Z k m , k i=1 1−q mi 1−q i , C k+1 ) exhibits the cyclic sieving phenomenon. arXiv:1207.5240v1 [math.CO]
European Journal of Combinatorics, 2012
We present an equivariant bijection between two actions-promotion and rowmotionon order ideals in... more We present an equivariant bijection between two actions-promotion and rowmotionon 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. Wieland's gyration. Finally, we define two actions with related orders on alternating sign matrices and totally symmetric self-complementary plane partitions.
The Electronic Journal of Combinatorics, May 30, 2013
Each positive rational number x>0 can be written uniquely as x=a/(b-a) for coprime positive integ... more Each positive rational number x>0 can be written uniquely as x=a/(b-a) for coprime positive integers 0<a<b. We will identify x with the pair (a,b). In this paper we define for each positive rational x>0 a simplicial complex \Ass(x)=\Ass(a,b) called the {\sf rational associahedron}. It is a pure simplicial complex of dimension a-2, and its maximal faces are counted by the {\sf rational Catalan number} \Cat(x)=\Cat(a,b):=\frac{(a+b-1)!}{a!\,b!}. The cases (a,b)=(n,n+1) and (a,b)=(n,kn+1) recover the classical associahedron and its "Fuss-Catalan" generalization studied by Athanasiadis-Tzanaki and Fomin-Reading. We prove that \Ass(a,b) is shellable and give nice product formulas for its h-vector (the {\sf rational Narayana numbers}) and f-vector (the {\sf rational Kirkman numbers}). We define \Ass(a,b) via {\sf rational Dyck paths}: lattice paths from (0,0) to (b,a) staying above the line y = \frac{a}{b}x. We also use rational Dyck paths to define a rational generalization of noncrossing perfect matchings of [2n]. In the case (a,b) = (n, mn+1), our construction produces the noncrossing partitions of [(m+1)n] in which each block has size m+1.
Journal of Combinatorial Theory, Series A, 2013
The main objects of noncrossing Catalan combinatorics associated to a finite Coxeter system are n... more The main objects of noncrossing Catalan combinatorics associated to a finite Coxeter system are noncrossing partitions, sortable elements, and cluster complexes. The first and the third of these have known Fuss-Catalan generalizations, for which we provide new viewpoints. We introduce a corresponding generalization of sortable elements as elements in the positive Artin monoid, and show how this perspective ties together all three generalizations.
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Papers by Nathan Williams