arXiv subjects

Per Alexandersson

Publications and source records attributed to Per Alexandersson.

At least 19 recordsLinked to original sources

arXiv2026

Sink polynomials of line graphs and weighted matching polynomials

Chromatic quasisymmetric functions and unicellular LLT polynomials give symmetric-function interpretations of graph orientations counted by sinks. For natural unit interval graphs, we prove that the acyclic sink polynomial is real-rooted for every nonnegative ascent weight. Equivalently, the elementary coefficients of the chromatic quasisymmetric function, grouped by partition length, form a real-rooted polynomial. The Carlsson--Mellit relation expresses this polynomial as a plethystic specialization of the corresponding LLT polynomial, and the Hessenberg correspondence gives an interpretation in cohomology. More generally, the acyclic sink polynomial is real-rooted for chordal claw-free graphs, and the all-orientations sink polynomial is real-rooted for every claw-free graph. We also study a real-rooted polynomial counting joint minima in local edge orders. It agrees with the acyclic sink polynomial of the line graph exactly for forests. An exact formula for unicyclic roots gives a line-graph counterexample to acyclic sink real-rootedness.

math.CO↗

arXiv2026

Polynomiality of Stretched Schubert Structure Constants and Key Coefficients

We prove that monomial coefficients in affine families of key and Schubert polynomials are eventually polynomial. These results extend to finite products. Using M.~Watanabe's Schubert duality, we deduce that stretched Schubert structure constants are eventually polynomial. This proves a conjecture stated by I.~Pak and Z.~Slonim. We also prove that the positive key expansion of every affine-code family of Schubert polynomials eventually has fixed coefficients and affine indices. The proofs combine Demazure operators and vector partition functions with P.~Magyar's orthodontic formula and the generalized Demazure character theory of V.~Lakshmibai, P.~Littelmann, and P.~Magyar.

math.CO↗

arXiv2026

Real-rooted Eulerian polynomials from permutations, words, and paths

We study seven Eulerian-type polynomial families. We prove that the descent polynomials of derangements are real-rooted, settling the derangement part of a conjecture of S.~Fu, Z.~Lin, and J.~Zeng. We also resolve the real-rootedness conjecture in OEIS A335340, strengthen the known rowwise real-rootedness of an even-top descent family to consecutive strict interlacing, and prove real-rootedness, consecutive interlacing, and real-rooted gamma-polynomials for U.~Shankar's super-Eulerian polynomials. A differential recurrence gives consecutive weak interlacing for ternary words counted by increasing runs. We prove stability of the peak-value refinement and consecutive interleaving of its positive weighted diagonals, settling a conjecture of P.~Alexandersson and O.~Nabawanda. Finally, we prove that the zig-zag Eulerian polynomials, which enumerate swaps in alternating permutations, have simple negative roots, settling the real-rootedness conjecture of T.~Kyle Petersen and Y.~Zhuang.

math.CO↗

arXiv2026

The e-positivity of $\mathrm{rowDiv}_k(e_μ)$

We prove that the partition division operator $\mathrm{rowDiv}_k$ preserves e-positivity, resolving a conjecture of Alexandersson and Dai. For $μ\vdash kn$ and $ν\vdash n$, we express the coefficient of $e_ν$ in $\mathrm{rowDiv}_k(e_μ)$ as a normalized sum of positive rational weights over closed walks on a finite directed graph of $k$-restricted partitions. The proof combines the Pieri rule and a root of unity factorization for Macdonald polynomials. As a corollary, we recover the h-positivity of the symmetric functions $A_{n,k}$ of Amdeberhan, Shareshian, and Stanley.

math.CO↗

arXiv2026

Flip classes and elementary expansions of vertical-strip LLT polynomials

After the substitution $q\mapsto q+1$, vertical-strip LLT polynomials are weighted sums of strict $P$-partition generating functions indexed by orientations of decorated unit-interval graphs, with weight $q^{\mathrm{asc}(θ)}$. A flip moves an ascending edge across a triangle while keeping its long edge. We show that the sum over every flip class is elementary-positive: it equals the sum of the elementary symmetric functions given by the highest reachable vertices of its orientations. This answers Hwang's question on the Schur positivity of LLT flip-class sums, and it holds for circular unit-interval graphs as well. Summing over classes gives a new proof of the elementary expansion of Alexandersson and Sulzgruber, and its circular analog, which was open. We also obtain a flip-class refinement of the elementary expansion of LLT cumulants.

math.CO↗

arXiv2026

Peaks and peak-nestings on unit interval graphs

Unit interval graphs admit a classical Catalan encoding by area sequences, or equivalently by Dyck paths. We introduce a new statistic on these graphs, called peak-nesting, defined as the largest number of peak-cliques containing a common vertex. Through the correspondence with area sequences, peak-nesting also defines a new statistic on Dyck paths. We construct a bijection between Dyck paths and bicolored Motzkin paths which simultaneously records peak-nesting and the number of peaks. This yields a refinement of Touchard's identity, coefficient formulas involving Dyck paths of given height, and new combinatorial interpretations for several sequences recorded in the OEIS. We study the peak and peak-nesting polynomials over all unit interval graphs and over the subclasses of connected, Abelian, $2$-nested, and reduced unit interval graphs. For these families we obtain closed formulas, recurrences, and, for the symmetric cases, nonnegative expansions in the gamma-basis with explicit combinatorial interpretations. We also investigate questions regarding the location of zeros and the log-concavity of the corresponding coefficient sequences: some of the polynomial families form generalized Sturm sequences, whereas others fail to be real-rooted but appear nevertheless to have log-concave coefficients. Finally, we relate the peak-clique presentation of a unit interval graph to lattice path matroids.

math.CO↗

arXiv2026

Real-rootedness and interlacing for parking functions and Chow polynomials

We prove real-rootedness for the Chow polynomials of the noncrossing partition lattices by transferring tieless parking functions to finite-alphabet Smirnov words and applying an interlacing-preserving transition of M.~Leander. We also derive a triangular recurrence for peaks and ties and identify the peakless-tieless descent polynomial as the Narayana polynomial. For the toric $g$-contributions of Ehrenborg--Hetyei--Readdy, we exhibit a fixed-row common interlacer and establish real-rootedness of all nonnegative row sums. Individual real-rootedness follows in particular; Q.~Xiao also proved it by a different differential recurrence. We also give a second proof of the individual statement, by finite Schur--Szegő convolution, that does not use the common interlacer. These results prove Conjecture~4.2 of Xiao and Conjecture~11.2 of Ehrenborg--Hetyei--Readdy, with consequences for weakly 123-avoiding parking functions. We also prove real-rootedness for the image-size polynomial on all parking functions and for the ascent and descent polynomials of four two-pattern-avoiding classes.

math.CO↗

arXiv2026

Some Families of Type $B$ Set Partitions Counted by the Dowling Numbers

In this paper, we study type $B$ set partitions without zero block. Certain classes of these partitions, such as merging-free and separated partitions (enumerated by the Dowling numbers), are investigated. We show that these classes are in bijection with type $B$ set partitions. The intersection of these two classes is also studied, and we prove that their block-generating polynomials are real-rooted. Finally, we study the descent statistics on the class of permutations obtained by flattening type $B$ merging-free partitions. Using the valley-hopping action, we prove the Gamma-positivity of the descent distribution and provide a combinatorial interpretation of the Gamma-coefficients. We also show that the descent statistic is homomesic under valley-hopping.

math.CO↗

arXiv2026

Rook matroids and log-concavity of $P$-Eulerian polynomials

We define and study rook matroids, the bases of which correspond to non-nesting rook placements on a skew Ferrers board. We show that rook matroids are a subclass of both transversal matroids and positroids; they also bear a subtle relationship to lattice path matroids that centers around not having the quaternary matroid $Q_{6}$ as a minor. The enumerative and distributional properties of non-nesting rook placements stand in contrast to those of usual rook placements: the non-nesting rook polynomial is not real-rooted in general, and is instead ultra-log-concave. We leverage this property together with a correspondence between rook placements and linear extensions of a poset to show that if $P$ is a naturally labeled width two poset, then the $P$-Eulerian polynomial $W_{P}$ is ultra-log-concave. This takes an important step towards resolving a log-concavity conjecture of Brenti (1989) and completes the story of the Neggers--Stanley conjecture for naturally labeled width two posets.

math.CO↗

arXiv2026

Fast computation of Ehrhart polynomials of Gelfand--Tsetlin polytopes via Macdonald reciprocity

We describe an efficient method for computing the Ehrhart polynomial of Gelfand--Tsetlin polytopes arising from Kostka coefficients. The key idea is to exploit Ehrhart--Macdonald reciprocity: evaluating the Ehrhart polynomial at negative integers reduces to counting \emph{strict} Gelfand--Tsetlin patterns, which are often zero or very small for low dilations. Combined with an adaptive strategy that chooses the cheapest evaluation point (positive or negative) at each step, this yields substantial practical speedups compared to general-purpose polytope software. We benchmark against $\mathtt{OSCAR}$/$\mathtt{polymake}$, and illustrate the broader applicability of the method through order polytopes and permutation posets. The implementation is available in the Rust \texttt{kostka} package, with related optimizations also incorporated in the new \texttt{lrcalc-rs} replacement for \texttt{lrcalc}.

math.CO↗

arXiv2026

On boundary points of minimal continuously Hutchinson invariant sets

A linear differential operator $T=Q(z)\frac{d}{dz}+P(z)$ with polynomial coefficients defines a continuous family of Hutchinson operators when acting on the space of positive powers of linear forms. In this context, $T$ has a unique minimal Hutchinson-invariant set $M_{CH}^{T}$ in the complex plane. Using a geometric interpretation of its boundary in terms of envelopes of certain families of rays, we subdivide this boundary into local and global arcs (the former being portions of integral curves of the rational vector field $\frac{Q(z)}{P(z)}\partial_{z}$), and singular points of different types which we classify below. The latter decomposition of the boundary of $M_{CH}^{T}$ is largely determined by its intersection with the plane algebraic curve formed by the inflection points of trajectories of the field $\frac{Q(z)}{P(z)}\partial_{z}$. We provide an upper bound for the number of local arcs in terms of degrees of $P$ and $Q$. As an application of our classification, we obtain a number of global geometric properties of minimal Hutchinson-invariant sets.

math.DS↗

arXiv2026

Partition division maps, symmetric functions and positivity

We study a linear map on symmetric functions that ``divides'' a partition by a positive integer $k$, sending a Schur function indexed by a partition of $kn$ to a symmetric function indexed by partitions of $n$. We determine its Schur expansion explicitly for Schur and skew Schur functions, showing that the coefficients are enumerated by a new family of combinatorial objects, called $k$-Yamanouchi tableaux, which generalize the classical ballot (Yamanouchi) tableaux appearing in the Littlewood--Richardson rule. We also study the images of elementary symmetric functions under this map, derive the power-sum expansion of their $ω$-images, and establish power-sum positivity. A further application establishes a connection to work of Tewodros Amdeberhan, John Shareshian, and Richard Stanley on alternating permutations and Euler numbers.

math.CO↗

arXiv2026

Polynomials from tilings of rectangles

We study tilings of rectangular boards using unit squares together with a single type of big tile shaped as a Ferrers diagram. We derive generating functions for these tilings, prove real-rootedness and interlacing properties of associated independence polynomials, and establish connections with several sequences in the OEIS. Our results touch on tilings involving L-shaped polyominoes, fault-free tilings, and cylindric variants. We prove that tiling polynomials for two-column Ferrers shapes are real-rooted and form interlacing sequences.

math.CO↗

arXiv2026

Decomposing Determinantal Varieties from Statistics via Matroid Theory

We study determinantal varieties from conditional independence models with hidden variables, focusing on their irreducible decompositions, dimensions, degrees, and Gröbner bases. Each variety encodes a collection of matroids, whose flats capture algebraic dependencies among variables. Using this approach, we provide a systematic description of the components, their dimensions, and defining equations, and introduce a combinatorial framework for computing the degree of the determinantal variety. Our approach highlights the central role of matroidal structures in the study of determinantal varieties and extends beyond the reach of current computational techniques.

math.CO↗

arXiv2025

A weighted Murnaghan-Nakayama rule for $(P, w)$-partitions

The $(P, w)$-partition generating function $K_{(P,w)}(x)$ is a quasisymmetric function obtained from a labeled poset. Recently, Liu and Weselcouch gave a formula for the coefficients of $K_{(P,w)}(x)$ when expanded in the quasisymmetric power sum function basis. This formula generalizes the classical Murnaghan--Nakayama rule for Schur functions. We extend this result to weighted $(P, w)$-partitions and provide a short combinatorial proof, avoiding the Hopf algebra machinery used by Liu--Weselcouch.

math.CO↗

arXiv2025

Real-rootedness of rook-Eulerian polynomials

We introduce rook-Eulerian polynomials, a generalization of the classical Eulerian polynomials arising from complete rook placements on Ferrers boards, and prove that they are real-rooted. We show that a natural context in which to interpret these rook placements is as lower intervals of $312$-avoiding permutations in the Bruhat order. We end with some variations and generalizations along this theme.

math.CO↗

arXiv2024

An inverse problem in Pólya--Schur theory. II. Exactly solvable operators and complex dynamics

This paper, being the sequel of [An inverse problem in Polya-Schur theory. I. Non-genegerate and degenerate operators], studies a class of linear ordinary differential operators with polynomial coefficients called \emph{exactly solvable}; such an operator sends every polynomial of sufficiently large degree to a polynomial of the same degree. We focus on invariant subsets of the complex plane for such operators when their action is restricted to polynomials of a fixed degree and discover a connection between this topic and classical complex dynamics and its multi-valued counterpart. As a very special case of invariant sets we recover the Julia sets of rational functions.

math.DS↗

arXiv2024

Linear first order differential operators and their Hutchinson-invariant sets

In this paper, we initiate the study of a new interrelation between linear ordinary differential operators and complex dynamics which we discuss in details in the simplest case of operators of order $1$. Namely, assuming that such an operator $T$ has polynomial coefficients, we interpret it as a continuous family of Hutchinson operators acting on the space of positive powers of linear forms. Using this interpretation of $T$, we introduce its continuously Hutchinson invariant subsets of the complex plane and investigate a variety of their properties. In particular, we prove that for any $T$ with non-constant coefficients, there exists a unique minimal under inclusion invariant set $\mathrm{M}^T_{CH}$ and find explixitly when it equals $\mathbb{C}$.

math.DS↗