Related papers: Set-Valued Young Tableaux and Product-Coproduct Pr…
Banderier, Marchal, and Wallner considered Young tableaux with walls, which are similar to standard Young tableaux, except that local decreases are allowed at some walls. In this work, we prove a conjecture of Fuchs and Yu concerning the…
We study root-theoretic Young diagrams to investigate the existence of a Lie-type uniform and nonnegative combinatorial rule for Schubert calculus. We provide formulas for (co)adjoint varieties of classical Lie type. This is a simplest case…
This is the first of three papers that develop structures which are counted by a "parabolic" generalization of Catalan numbers. Fix a subset R of {1,..,n-1}. Consider the ordered partitions of {1,..,n} whose block sizes are determined by R.…
The purpose of this paper is twofold. First, we present a conjecture to the effect that the ranks of the syzygy modules of a smooth projective variety become normally distributed as the positivity of the embedding line bundle grows. Then,…
For the vacuum sectors of regime-III ABF models, we observe that two sets of combinatorial objects: the strings which parametrize the row-to-row transfer matrix eigenvectors, and the paths which parametrize the corner transfer matrix…
Young's lattice is a partial order on integer partitions whose saturated chains correspond to standard Young tableaux, one type of combinatorial object that generates the Schur basis for symmetric functions. Generalizing Young's lattice, we…
Schur-Weyl duality is a fundamental framework in combinatorial representation theory. It intimately relates the irreducible representations of a group to the irreducible representations of its centralizer algebra. We investigate the analog…
We consider $\beta$--Plancherel measures \cite{Ba.Ra.} on subsets of partitions -- and their asymptotics. These subsets are the Young diagrams contained in a $(k,\ell)$--hook, and we calculate the asymptotics of the expected shape of these…
We prove a new CLT for the difference of linear eigenvalue statistics of a Wigner random matrix $H$ and its minor $\hat H$ and find that the fluctuation is much smaller than the fluctuations of the individual linear statistics, as a…
Usual math sets have special types: countable, compact, open, occasionally Borel, rarely projective, etc. Each such set is described by a single Set Theory formula with parameters unrelated to other formulas. Exotic expressions involving…
In this thesis we enumerate standard young tableaux (SYT) of certain truncated skew shapes, which we call battery shapes. This is motivated by a chess problem. In an enumerative chess problem, the set of moves in the solution is (usually)…
In how many ways can you place n chocolate pieces all of different sizes in an n by n chocolate box, in such a way that when you go from left to right and from top to bottom, there are no gaps AND the sizes increase along each row and each…
We give a crystal structure on the set of Gelfand-Tsetlin patterns which parametrize bases for finite-dimensional irreducible representations of the general linear Lie algebra. The crystal data are given in closed form, expressed using…
This paper studies planar algebras of Jones' style associated with the Young graph. We first see that, given a positive real valued function on the Young graph, we may obtain a planar algebra whose structure is defined in terms of a state…
We revisit tensor algebras of subproduct systems with Hilbert space fibers, resolving some open questions in the case of infinite dimensional fibers. We characterize when a tensor algebra can be identified as the algebra of uniformly…
Let s,t,m,n be positive integers such that sm=tn. Let M(m,s;n,t) be the number of m x n matrices over {0,1,2,...} with each row summing to s and each column summing to t. Equivalently, M(m,s;n,t) counts 2-way contingency tables of order m x…
In our joint paper with W. Fulton (math.AG/9804041) we prove a formula for the cohomology class of a quiver variety. This formula involves a new class of generalized Littlewood-Richardson coefficients, all of which surprisingly seem to be…
Three-dimensional random tessellations that are stable under iteration (STIT tessellations) are considered. They arise as a result of subsequent cell division, which implies that their cells are not face-to-face. The edges of the…
Starting from a succession rule for Catalan numbers, we define a procedure encoding and listing the objects enumerated by these numbers such that two consecutive codes of the list differ only for one digit. Gray code we obtain can be…
Work of Gaetz, Pechenik, Pfannerer, Striker, and Swanson (2024) introduced promotion permutations for a rectangular standard Young tableau $T$. These promotion permutations encode important features of $T$ and its orbit under…