arXiv subjects

Zachary Hamaker

Publications and source records attributed to Zachary Hamaker.

At least 19 recordsLinked to original sources

arXiv2026

Coxeter and Schubert combinatorics of $\mu$-Involutions

The variety of complete quadrics is the wonderful compactification of $GL_n/O_n$ and admits a cell decomposition into Borel orbits indexed by combinatorial objects called $\mu$-involutions. We study Coxeter-theoretic properties of $\mu$-involutions with results including a combinatorial description for their atoms, an exchange lemma, and transposition-like operators that characterize their Bruhat order. The corresponding orbit closures can be realized inside the flag variety. In this setting, we study the cohomology representatives of these orbits, which are, up to a scalar, the $\mu$-involution Schubert polynomials. We expand $\mu$-involution Schubert polynomials as a multiplicity-free sum of $\nu$-involution Schubert polynomials when $\nu$ refines $\mu$ and provide recurrences analogous to Monk's rule for Schubert polynomials.

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arXiv2025

Odd Shifted Parking Functions

Stanley recently introduced the shifted parking function symmetric function $SH_n$, which is the shiftification of Haiman's parking function symmetric function $PF_n$. The function $SH_n$ lives in the subalgebra of symmetric functions generated by odd power sums. Stanley showed how to expand $SH_n$ into the $V-$basis of this algebra, which is indexed by partitions with all parts odd and is analogous to the complete homogeneous (or elementary) basis of symmetric functions. We introduce odd shifted parking functions to give combinatorial and representation-theoretic realizations of the $V-$expansion of $SH_n$, resolving the main open problem in his paper. Further, we present two representation-theoretic realizations of shiftification allowing us to interpret $SH_n$ as the spin character of a projective representation. We conclude with further directions, including a relationship between $SH_n$ and Haglund's $(q,t)-$Schr\"oder theorem.

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arXiv2025

Partial permutations and character evaluations

Let $I = (i_1, \dots, i_k)$ and $J = (j_1, \dots, j_k)$ be two length $k$ sequences drawn from $\{1, \dots, n \}$. We have the group algebra element $[I,J] := \sum_{w(I) = J} w \in \mathbb{C}[\mathfrak{S}_n]$ where the sum is over permutations $w \in \mathfrak{S}_n$ which satisfy $w(i_p) = j_p$ for $p = 1, \dots, k$. We give an algorithm for evaluating irreducible characters $\chi^\lambda: \mathbb{C}[\mathfrak{S}_n] \to \mathbb{C}$ of the symmetric group on the elements $[I,J]$. This algorithm is a hybrid of the classical Murnaghan--Nakayama rule and a new path Murnaghan--Nakayama rule which reflects the decomposition of a partial permutation into paths and cycles. These results first appeared in arXiv:2206.06567, which is no longer intended for publication. We originally used the character theoretic results in this paper to prove asymptotic results on moments of certain permutation statistics restricted to conjugacy classes. A referee generously shared a combinatorial argument which is strong enough to prove these results without recourse to character theory. These results now appear in our companion paper~\cite{HRMoment}. However, the approach in this paper is more explicit, as we demonstrate with several examples.

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arXiv2025

Moments of permutation statistics by cycle type

Beginning with work of Zeilberger on classical pattern counts, there are a variety of structural results for moments of permutation statistics applied to random permutations. Using tools from representation theory, Gaetz and Ryba generalized Zeilberger's results to uniformly random permutations of a given cycle type. We introduce regular statistics and characterize their moments for all cycle types, generalizing all results in this literature that we are aware of. Our approach splits into two steps: first characterize such statistics as linear combinations of indicator functions for partial permutations, then identifying the moments of such indicators. As an application, we show that many regular statistics exhibit a law of large numbers depending only on fixed point counts and a similar variance property that depends also on two--cycle counts. These results first appeared in arXiv:2206.06567, which is no longer intended for publication. Our original proof of the moment result for indicators of partial permutations relied on representation theory and symmetric functions. A referee generously shared a combinatorial argument, allowing us to give a self-contained treatment of these results that does not rely on representation theory.

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arXiv2025

Type C $K$-Stanley symmetric functions and Kra\'skiewicz-Hecke insertion

We study Type C $K$-Stanley symmetric functions, which are $K$-theoretic extensions of the Type C Stanley symmetric functions. They are indexed by signed permutations and can be used to enumerate reduced words via their expansion into Schur $Q$-functions, which are indexed by strict partitions. A combinatorial description of the Schur $Q$- coefficients is given by Kra\'skiewicz insertion. Similarly, their $K$-Stanley analogues are conjectured to expand positively into $GQ$'s, which are $K$-theory representatives for the Lagrangian Grassmannian introduced by Ikeda and Naruse also indexed by strict partitions. We introduce a $K$-theoretic analogue of Kra\'skiewicz insertion, which can be used to enumerate 0-Hecke expressions for signed permutations and gives a conjectural combinatorial rule for computing this $GQ$ expansion. We show the Type C $K$-Stanleys for certain fully commutative signed permutations are skew $GQ$'s. Combined with a Pfaffian formula of Anderson's, this allows us to prove Lewis and Marberg's conjecture that $GQ$'s of (skew) rectangle shape are $GQ$'s of trapezoid shape. Combined with our previous conjecture, this also gives an explicit combinatorial description of the skew $GQ$ expansion into $GQ$'s. As a consequence, we obtain a conjecture for the product of two $GQ$ functions where one has trapezoid shape.

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arXiv2024

Tableau formula for vexillary double Edelman--Greene coefficients

Lam, Lee and Shimozono recently introduced backstable double Grothendieck polynomials to represent $K$-theory classes of the infinite flag variety. They used them to define double $β$-Stanley symmetric functions, which expand into double stable Grothendieck functions with polynomial coefficients called double $β$-Edelman--Greene coefficients. Anderson proved these coefficients are $β$-Graham positive. For vexillary permutations, this is equivalent to a statement for skew flagged double $β$-Grothendieck functions. Working in this setting, we give a tableau formula for vexillary double $β$-Edelman--Greene coefficients that is manifestly $β$-Graham positive. Our formula demonstrates a finer notion of positivity than was previously known.

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arXiv2024

Characters of local and regular permutation statistics

The goal of this monograph is to study the indicator function for a set of permutations mapping one finite sequence of positive integers to another from a representation theoretic, combinatorial and probabilistic perspective. The degree of a function of permutations is the size of the largest pair of sequences required when expressing it as a linear combination of these indicators. This notion of degree, implicit in work of Diaconis, is critical for many applications of representation theory to extremal combinatorics, machine learning, probability and statistics. We use the term local to indicate bounded degree and initiate the study of low degree class functions, which encode probabilistic data for permutation statistics on each cycle type simultaneously. We begin with a self contained treatment of for functions of permutations, developing its theory using language familiar to enumerative and algebraic combinatorialists. This leads naturally to a novel basis for symmetric functions we call the path power sum symmetric functions. The most technically challenging part of our work is the path Murnaghan-Nakayama formula, which expands path power sums into Schur functions. By combining the the path Murnaghan-Nakayam formula with the classical theory of character polynomials, one obtains a structural characterization for moments for permutation statistics conditioning on cycle type. We then analyze asymptotic properties of these moments. In doing so, we introduce the novel family of regular permutation statistics, which include almost all reasonable weighted pattern counting statistics. We show a large family of regular statistics satisfy a law of large numbers on a given cycle type depending only on the proportion of fixed points and a have variances depending only on fixed points and two cycles.

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arXiv2023

Dual Schubert polynomials via a Cauchy identity

We give a combinatorial proof that Postnikov and Stanley's formula for dual Schubert polynomials in terms of weighted chains in Bruhat order is equivalent to a classical Cauchy identity for polynomials. This gives a natural interpretation of Huang and Pylyavskyy's recent insertion algorithms.

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arXiv2022

Three coloring via triangle counting

In the first partial result toward Steinberg's now-disproved three coloring conjecture, Abbott and Zhou used a counting argument to show that every planar graph without cycles of lengths 4 through 11 is 3-colorable. Implicit in their proof is a fact about plane graphs: in any plane graph of minimum degree 3, if no two triangles share an edge, then triangles make up strictly less than 2/3 of the faces. We show how this result, combined with Kostochka and Yancey's resolution of Ore's conjecture for k = 4, implies that every planar graph without cycles of lengths 4 through 8 is 3-colorable.

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arXiv2022

Lenart's bijection via bumpless pipe dreams

Pipe dreams and bumpless pipe dreams for vexillary permutations are each known to be in bijection with certain semistandard tableaux via maps due to Lenart and Weigandt, respectively. Recently, Gao and Huang have defined a bijection between the former two sets. In this note we show for vexillary permutations that the Gao-Huang bijection preserves the associated tableaux, giving a new proof of Lenart's result. Our methods extend to give a recording tableau for any bumpless pipe dream.

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arXiv2022

On pattern avoidance in matchings and involutions

We study the relationship between two notions of pattern avoidance for involutions in the symmetric group and their restriction to fixed-point-free involutions. The first is classical, while the second appears in the geometry of certain spherical varieties and generalizes the notion of pattern avoidance for perfect matchings studied by Jelínek. The first notion can always be expressed in terms of the second, and we give an effective algorithm to do so. We also give partial results characterizing the families of involutions where the converse holds. As a consequence, we prove two conjectures of McGovern characterizing (rational) smoothness of certain varieties. We also give new enumerative results, and conclude by proposing several lines of inquiry that extend our current work.

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arXiv2021

Involution pipe dreams

Involution Schubert polynomials represent cohomology classes of $K$-orbit closures in the complete flag variety, where $K$ is the orthogonal or symplectic group. We show they also represent $T$-equivariant cohomology classes of subvarieties defined by upper-left rank conditions in the spaces of symmetric or skew-symmetric matrices. This geometry implies that these polynomials are positive combinations of monomials in the variables $x_i + x_j$, and we give explicit formulas of this kind as sums over new objects called involution pipe dreams. Our formulas are analogues of the Billey-Jockusch-Stanley formula for Schubert polynomials. In Knutson and Miller's approach to matrix Schubert varieties, pipe dream formulas reflect Gröbner degenerations of the ideals of those varieties, and we conjecturally identify analogous degenerations in our setting.

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arXiv2021

Involutions under Bruhat order and labeled Motzkin Paths

In this note, we introduce a statistic on Motzkin paths that describes the rank generating function of Bruhat order for involutions. Our proof relies on a bijection introduced by Philippe Biane from permutations to certain labeled Motzkin paths and a recently introduced interpretation of this rank generating function in terms of visible inversions. By restricting our identity to fixed-point-free (FPF) involutions, we recover an identity due to Louis Billera, Lionel Levine and Karola Mészáros with a previous bijective proof by Matthew Watson. Our work sheds new light on the Ethiopian dinner game.

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arXiv2021

Bijecting hidden symmetries for skew staircase shapes

We present a bijection between the set of standard Young tableaux of staircase minus rectangle shape, and the set of marked shifted standard Young tableaux of a certain shifted shape. Numerically, this result is due to DeWitt (2012). Combined with other known bijections this gives a bijective proof of the product formula for the number of standard Young tableaux of staircase minus rectangle shape. This resolves an open problem by Morales, Pak and Panova (2019), and allows for efficient random sampling. Other applications include a bijection for semistandard Young tableaux, and a bijective proof of Stembridge's symmetry of LR-coefficients of the staircase shape. We also extend these results to set-valued standard Young tableaux in the combinatorics of K-theory, leading to new proofs of results by Lewis and Marberg (2019) and Abney-McPeek, An and Ng (2020).

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arXiv2020

Gröbner geometry of Schubert polynomials through ice

The geometric naturality of Schubert polynomials and their combinatorial pipe dream representations was established by Knutson and Miller (2005) via antidiagonal Gröbner degeneration of matrix Schubert varieties. We consider instead diagonal Gröbner degenerations. In this dual setting, Knutson, Miller, and Yong (2009) obtained alternative combinatorics for the class of "vexillary'' matrix Schubert varieties. We initiate a study of general diagonal degenerations, relating them to a neglected formula of Lascoux (2002) in terms of the $6$-vertex ice model (recently rediscovered by Lam, Lee, and Shimozono (2018) in the guise of "bumpless pipe dreams'').

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arXiv2019

Wronskian Appell Polynomials and Symmetric Functions

We study Wronskians of Appell polynomials indexed by integer partitions. These families of polynomials appear in rational solutions of certain Painlevé equations and in the study of exceptional orthogonal polynomials. We determine their derivatives, their average and variance with respect to Plancherel measure, and introduce several recurrence relations. In addition, we prove an integrality conjecture for Wronskian Hermite polynomials previously made by the first and last authors. Our proofs all exploit strong connections with the theory of symmetric functions.

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arXiv2019

Weak order and descents for monotone triangles

Monotone triangles are a rich extension of permutations that biject with alternating sign matrices. The notions of weak order and descent sets for permutations are generalized here to monotone triangles, and shown to enjoy many analogous properties. It is shown that any linear extension of the weak order gives rise to a shelling order on a poset, recently introduced by Terwilliger, whose maximal chains biject with monotone triangles; among these shellings are a family of EL-shellings. The weak order turns out to encode an action of the 0-Hecke monoid of type A on the monotone triangles, generalizing the usual bubble-sorting action on permutations. It also leads to a notion of descent set for monotone triangles, having another natural property: the surjective algebra map from the Malvenuto- Reutenauer Hopf algebra of permutations into quasisymmetric functions extends in a natural way to an algebra map out of the recently-defined Cheballah-Giraudo-Maurice algebra of alternating sign matrices.

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arXiv2018

Pattern avoidance and quasisymmetric functions

Given a set of permutations Pi, let S_n(Pi) denote the set of permutations in the symmetric group S_n that avoid every element of Pi in the sense of pattern avoidance. Given a subset S of {1,...,n-1}, let F_S be the fundamental quasisymmetric function indexed by S. Our object of study is the generating function Q_n(Pi) = sum F_{Des sigma} where the sum is over all sigma in S_n(Pi) and Des sigma is the descent set of sigma. We characterize those Pi contained in S_3 such that Q_n(Pi) is symmetric or Schur nonnegative for all n. In the process, we show how each of the resulting Pi can be obtained from a theorem or conjecture involving more general sets of patterns. In particular, we prove results concerning symmetries, shuffles, and Knuth classes, as well as pointing out a relationship with the arc permutations of Elizalde and Roichman. Various conjectures and questions are mentioned throughout.

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