Talks
Title: Some Catalan Musings
Abstract: The sequence 1, 1, 2, 5, 14, 42, ... of Catalan numbers is probably
the most ubiquitous and most interesting sequence in
combinatorics. For instance, the Online Encyclopedia of Integer
Sequences says that the Catlan number entry A000108 "is probably the
longest entry in the OEIS, and rightly so." I have recently completed
a monograph on Catalan numbers, to be published by Cambridge
University Press. I will give a sample of results from this monograph,
including selections from the history of Catalan numbers, their
"transparent" combinatorial interpretations, more subtle combinatorial
interpretations, algebraic interpretations, and analytic and
number-theoretic properties.
Title: On the free Lie algebra with multiple brackets
Abstract: It is a classical result that the multilinear component of the free Lie algebra Lie(n) is isomorphic as a representation of the symmetric group to the top cohomology of the poset of partitions of an n-set tensored with the sign representation. Hence, the algebraic object Lie(n) can be understood by applying poset theoretic techniques to the poset of partitions whose description is purely combinatorial. We generalize this result in order to study the multilinear component of the free Lie algebra with multiple compatible Lie brackets. A new poset of weighted partitions, that generalizes the one introduced by Dotsenko and Khoroshkin, allows us to generalize the result. In particular we obtain combinatorial bases and compute the dimensions of these modules answering a question posed by Liu.
Title: Subtraction-free computations
Abstract: I will discuss some algorithms for efficient computation of generating functions (e.g., Schur polynomials) which do not use subtraction. Some of these algorithms do not use division either.
Title: Discrete homotopy and homology theory for metric spaces
Abstract: Discrete homotopy theory is a discrete analogue of homotopy theory, associating a bigraded sequence of groups to a simplicial complex, capturing its combinatorial structure, rather than its topological structure. Discrete homotopy can be equivalently defined for finite connected graphs, resulting in an algebraic invariant of finite connected graphs and graphs homomorphisms, in the same way that classical homotopy theory gives invariants of topological spaces and continuous maps: it associates a sequence of groups to a finite connected graph.The notion of discrete homotopy can also be generalized to arbitrary metric spaces. A natural question is whether one can define a corresponding notion of discrete homology in the same way that classical homology is related to classical homotopy theory. The positive answer to this question is the subject of this talk.
Title: Some of my joint work with Michelle Wachs
Abstract: I will give a survey some past and present work with Michelle on complexes of graphs and hypergraphs, and on quasisymmetric functions.
Title: Pure and nonpure combinatorics
Abstract: My collaboration with Michelle dates back 35 years. In this talk I will reminisce about
some of our joint work, in particular concerning nonpure complexes. I will end by presenting recent work, joint with Karim Adiprasito and Afshin Goodarzi, characterizing h-triangles of sequentially Cohen-Macaulay complexes. This answers a question posed by Michelle and myself.
Title: Generalized stability of Kronecker coefficients
Abstract: Kronecker coefficients are tensor product multiplicities for symmetric
group representations. It is well-known that not much is known about
them. In this talk we plan to discuss some new theorems and conjectures
about how these coefficients stabilize or de-stabilize in various
limiting cases. A typical (easy) case is the classical result of
Murnaghan that corresponds to incrementing the first rows of a triple
of Young diagrams.
Title: On monotonizable set partitions and separated subsequences
Abstract: It follows from a famous result of Erdos and Szekeres that every sequence of f(n) = n2 - 2n + 2 distinct numbers has a monotonic subsequence of length n, and that f(n) is the least number with this property. I will present an analogue in the context of set partitions. Specifically, I will show that any set partition of a set of g(n) = ⌊(n+1)2/4⌋ numbers has a monotonizable subpartition of weight n, and g(n) is the least number with this property. A well-known bijection translates this result to sequences: every sequence of length g(n) has a separated subsequence of length n, and g(n) is the least number with this property.
The set of monotonizable subpartitions of a given partition is a hereditary set. Thus, it is possible to study the topology of the abstract simplicial complex of monotonizable subpartitions of a given partition. I will present some preliminary observations about these objects.
This is joint work with Mike Sheard.
Title: Representation stability in the partition lattice
Abstract: There has been a wealth of work done over several decades regarding Sn-module structure for the rank-selected homology and Whitney homology of the partition lattice by Hanlon, Stanley, Sundaram, Wachs, and numerous others. This talk will focus on new results and new questions that came out of taking a representation theoretic stability perspective. This is joint work with Vic Reiner.
Title: Rook theory and simplicial complexes
Abstract: Rook theory deals with placements of nonattacking rooks on a board -- a subset of [n] x [n], where [n]={1,2,..., n}. The rook numbers of a board count placements of k nonattacking rooks on the board. The hit numbers of the board count placements of n nonattacking rooks on [n] x [n] in which k of the rooks lie on the board. The fundamental identity of rook theory relates the rook numbers and hit numbers of a board.
The sets of nonattacking rook positions in [n] x [n] form a pure simplicial complex with the property that any two faces of the same size are contained in the same number of facets, and this property is all we need to prove the fundamental identity. Thus we can generalize the fundamental identity to other simplicial complexes with the same property.
We will study analogues in this context of the factorial rook polynomial, introduced by Goldman, Joichi, and White, and of its reciprocity theorem, which relates the rook numbers of a board to the rook numbers of a complementary board. In particular, we will see how the reciprocity theorem of Bedrosian and Kelmans for spanning tree polynomials is part of this theory.
Title: The Combinatorics of the Hook case of the Shuffle Conjecture and its extensions
Abstract: The hook case of the character of diagonal harmonics is known as the q,t-Schroder polynomial since it can be described as a weighted sum over Schroder paths. We discuss some of the combinatorial issues involved with this and also the hook case of recent extensions of the Shuffle Conjecture. Connections to the theory of the superpolynomial knot invariant are highlighted.
Title: TBA
Abstract: TBA
Title: A bijective proof of Macdonald's reduced word formula
Abstract: Macdonald gave a remarkable formula connecting a weighted sum of
reduced words for a permutation with the number of terms in a Schubert
polynomial. We give a bijective proof of this formula based on
Little's bumping algorithm. We will also discuss some generalizations
of this formula based on work of Fomin, Kirillov, Stanely and Wachs.
This project extends earlier work by Benjamin Young on a Markov
process for reduced words of the longest permutation.
This is joint work with Ben Young and Alexander Holroyd.
Title: Some thoughts on the cone of log-concavity
Abstract: Let {h1, h2, ... } be a set of algebraically independent variables. We ask which vectors are extreme in the cone generated by hi hj - hi+1 hj-1 (i ≥ j > 0)
and hi (i > 0). We call this cone the cone of log concavity. More generally, we ask which vectors are extreme in the cone generated by Schur functions of partitions with k or fewer parts.
Title: q-Narayana and q-Kreweras numbers for Weyl groups
Abstract: (joint work with E. Sommers)
Among other things, Catalan numbers count noncrossing partitions, while the Narayana and Kreweras numbers refine this to count the noncrossing partitions according to their total number of blocks, and their list of block sizes.
Work of Sommers on nilpotent orbits suggested a way to define q-analogues of Kreweras numbers for a Weyl group W, summing to the known q-Catalan numbers for W. These q-Kreweras numbers have lots of other good properties:
(1) They sum to give q-Narayana numbers that agree in types A, B with ones in the literature (including recent work of M. Wachs).
(2) They give a q-analogue of the h-to-f-vector transformation for cluster complexes of finite type.
(3) They exhibit cyclic sieving phenomena for W-noncrossing partitions,
at least in the classical types A,B,C,D, and conjecturally in all types.
We will define the q-Kreweras, q-Narayana, q-Catalan numbers, explain some of their properties, and then pose some questions about them.
Title: Maximal increasing sequences in fillings of almost-moon polyominoes
Abstract: It was known that the number of fillings with zeros and ones of a given moon polyomino that do not contain a northeast chain of size k depends only on the set of columns of the polyomino, but not the shape of the polyomino. In this talk we present a bijective proof for this result by considering fillings of almost-moon polyominoes, which are moon polyominoes after removing one of the rows. Explicitly,
we construct a bijection which preserves the size of the largest northeast chains of the fillings when two adjacent rows of the polyomino are exchanged. This bijection also preserves the column sum of the fillings. We also present a bijection that preserves the size of the largest northeast chains, the row sum and the column sum if every row of the fillings has at most one 1.
This is a joint work with Svetlana Poznanovic.
Title: Weighted enumeration of consecutive 123-avoiding permutations,
the Hurwitz zeta function and the derivatives of cotangent
Abstract: A permutation π=(π1,...,πn) is consecutive 123-avoiding if there is no index i such that πi < πi+1 < πi+2. Similarly, a permutation π is cyclically consecutive 123-avoiding if the indicies are viewed modulo n. We consider a weighted enumeration problem of consecutive 123-avoiding permutations, where we able to to
determine an asymptotic expansion of the weighted enumeration. For the cyclically weighted enumeration we are able to determine an exact expression in terms of the Hurwitz zeta function. This yields explicit combinatorial expression for the higher derivatives of the cotangent function.
Title: Orthogonal polynomials and the 2-species ASEP
Abstract: The asymmetric exclusion process (ASEP) is a model of particles hopping on a 1-dimensional lattice. The partition function of this model is related to moments of Askey-Wilson polynomials. Recently Eric Rains defined moments of Koornwinder polynomials at q=t, and conjectured they are positive when written appropriately in the parameters of the ASEP. I'll explain joint work with Sylvie Corteel in which we prove a special case of Rains' conjecture by relating
Koornwinder moments to the 2-species ASEP, an exclusion process
involving two types of particles. I'll also describe complementary work
of Olga Mandelshtam and Xavier Viennot providing a combinatorial
description of the stationary distribution of the 2-species ASEP.
Title: The q = -1 phenomenon via homology concentration
Abstract: A homological approach to exhibiting instances of Stembridge's q = -1 phenomenon is introduced. This approach is shown to explain two important instances of the phenomenon: partitions whose Ferrers diagrams fit in a rectangle of fixed size and plane partitions fitting in a box of fixed size. We show that the fixed points of an involution gives a basis for homology with coefficients taken mod 2. A more general framework of invariant and coinvariant complexes is developed. Conjectures for necklaces and related open questions will be discussed. This is joint work with Tricia Hersh and John Shareshian.
Title: Lessons from Michelle
Title: Negative q-Stirling numbers
Abstract: The notion of the negative q-binomial was recently introduced by Fu,
Reiner, Stanton and Thiem. Mirroring the negative q-binomial, we show
the classical q-Stirling numbers of the second kind can be expressed
as a pair of statistics on a subset of restricted growth words. The
resulting expressions are polynomials in q and 1+q. We extend this
enumerative result via a decomposition of the Stirling poset, as well
as a homological version of Stembridge's q = -1 phenomenon. A parallel
enumerative, poset theoretic and homological study for the q-Stirling
numbers of the first kind is done beginning with de Médicis and
Leroux's rook placement formulation. Letting t = 1+q we give a
bijective combinatorial argument a la Viennot showing the
(q,t)-Stirling numbers of the first and second kind are orthogonal.
This is joint work with Yue Cai.
Title: Symplectic meanders
Abstract: Seaweed subalgebras of the general linear Lie algebra were introduced by Dergachev and Kirillov. They associated a graph called a meander to each seaweed subalgebra, and showed that the index of the seaweed subalgebra is equal to the number of connected components plus the number of cycles of the meander. Panyushev generalized the notion of seaweed subalgebras to any semisimple Lie algebra, and gave an inductive formula for the index. For the symplectic Lie algebra, we introduce symplectic meanders. We show that the index of a symplectic seaweed subalgebra can be computed from the properties of the associated symplectic meander. We discuss some consequences of this association. This is joint work with Vince Coll, Colton Magnant, and Hua Wang.
Title: Symmetric Function Operators and Parking Functions
Abstract: The main result presented in this talk is a plethystic formula for the specialization at t = 1/q of the Qu,v operators studied in [Math ArXiv: 1405.0316]. This discovery yields elementary and direct derivations of several identities relating these operators at t = 1/q to the Rational Compositional Shuffle conjecture of [math arXiv:1404.4616]. In particular we are able to give a direct derivation of a simple formula for the symmetric polynomial Qkm,kn 1 for t = 1/q for all (m,n) co-prime and k ≥ 1. We also give an elementary proof that this polynomial is Schur positive. Moreover, by combining our main result with the Rational Compositional Shuffle conjecture, we obtain a completely elementary derivation of the identity expressing this polynomial in terms of Parking functions in the km x kn lattice rectangle. (This is joint work with E. Leven, N. Wallach and G. Xin.)