Supervised works

Consider a directed multigraph D that is balanced (i.e., at each vertex, the indegree equals the outdegree). Let A be its set of arcs. Fix an integer k. Let s be a vertex of D. We show that the number of k-element subsets B of A that contain no cycles but contain a path from each vertex to s (we call them "s-convergences") is independent of s. This generalizes known facts about spanning arborescences, acyclic orientations and maximal acyclic subdigraphs. Moreover, this result can be generalized even further, replacing "contain no cycles" with "have a given set of cycles".

We address two variants of the classical necklace counting problem from enumerative combinatorics. In both cases, we fix a finite group G and a positive integer n. In the first variant, we count the “identity-product n-necklaces” — that is, the orbits of n-tuples (a₁, a₂, ..., a_n) ∈ Gⁿ that satisfy a₁a₂...a_n = 1 under cyclic rotation.

In the second, we count the orbits of all n-tuples (a₁, a₂, ..., a_n) ∈ Gⁿ under cyclic rotation and left multiplication (i.e., the operation of G on Gⁿ given by h · (a₁, a₂, ..., a_n) = (ha₁, ha₂, ..., ha_n)). We prove bijectively that both answers are the same, and express them as a sum over divisors of n.

Consequently, we generalize the first problem to n-necklaces whose product of entries lies in a given subset of G (closed under conjugation), and we connect a particular case to the enumeration of irreducible polynomials over a finite field with given degree and second-highest coefficient 0.

Part of a student project mentored via the Yulia's Dream program 2022-2023.

We prove a formula for the image of a skew Schur polynomial s_λ/μ (x₁, x₂, ..., x_N) under the differential operator ∇ = ∂ / ∂x₁ + ∂ / ∂x₂ + ... + ∂ / ∂x_N. This generalizes a formula of Weigandt for ∇(s_λ).

Student project mentored via the Yulia's Dream program 2022-2023. (Proof found by the students, writing mostly by me.)

The classical hook length formula of enumerative combinatorics expresses the number of standard Young tableaux of a given partition shape as a single fraction. In recent years, two generalizations of this formula have emerged: one by Pak and Postnikov, replacing the number by a (rational) generating function, and one by Naruse, which generalizes the setting from a partition to a skew partition. Both generalizations appear to lie significantly deeper, with no simple proofs known. We combine them into a generating-function identity for skew partitions, and prove it in a fairly elementary way using recursion, determinants and simple combinatorics.

This is an expository note (new proof of an old result) resulting from a PRIMES reading project. See also A generalization of Chio Pivotal Condensation.

We find a new approach to computing the remainder of a polynomial modulo xⁿ - 1; such a computation is called modular enumeration. Given a polynomial with coefficients from a commutative ℚ-algebra, our first main result constructs the remainder simply from the coefficients of residues of the polynomial modulo Φ_d(x) for each d | n. Since such residues can often be found to have nice values, this simplifies a number of modular enumeration problems; indeed in some cases, such residues are already known while the related modular enumeration problem has remained unsolved. We list six such cases which our technique makes easy to solve. Our second main result is a formula for the unique polynomial a such that a ≡ f mod Φ_n(x) and a ≡ 0 mod x^d - 1 for each proper divisor d of n.

We find a formula for remainders of q-multinomial coefficients and for remainders of q-Catalan numbers modulo qⁿ - 1, reducing each problem to a finite number of cases for any fixed n. In the prior case, we solve an open problem posed by Hartke and Radcliffe. In considering q-Catalan numbers modulo qⁿ - 1, we discover a cyclic group operation on certain lattice paths which behaves predictably with regard to major index. We also make progress on a problem in modular enumeration on subset sums posed by Kitchloo and Pachter.

Cylindric Young tableaux are combinatorial objects that first appeared in the 1990s. A natural extension of the classical notion of a Young tableau, they have since been used several times, most notably by Gessel and Krattenthaler and by Alexander Postnikov. Despite this, relatively little is known about cylindric Young tableaux. This paper is an investigation of the properties of this object. In this paper, we extend the Robinson--Schensted--Knuth correspondence, a well-known and very useful bijection concerning regular Young tableaux, to be a correspondence between pairs of cylindric tableaux. We use this correspondence to reach further results about cylindric tableaux. We then establish an interpretation of cylindric tableaux in terms of a game involving marble-passing. Next, we demonstrate a generic method to use results concerning cylindric tableaux in order to prove results about skew Young tableaux. We finish with a note on Knuth equivalence and its analog for cylindric tableaux.

We study a family of equivalence relations on S_n, the group of permutations on n letters, created in a manner similar to that of the Knuth relation and the forgotten relation. For our purposes, two permutations are in the same equivalence class if one can be reached from the other through a series of pattern-replacements using patterns whose order permutations are in the same part of a predetermined partition of S_c.

When the partition is of S₃ and has one nontrivial part and that part is of size greater than two, we provide formulas for the number of classes created in each previously unsolved case. When the partition is of S₃ and has two nontrivial parts, each of size two (as do the Knuth and forgotten relations), we enumerate the classes for 13 of the 14 unresolved cases. In two of these cases, enumerations arise which are the same as those yielded by the Knuth and forgotten relations. The reasons for this phenomenon are still largely a mystery.