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Towards the classification of maximum scattered linear sets of PG(1, q^5)
Every maximum scattered linear set in PG(1, q5) is the projection of an Fq -subgeometry Σ of PG(4, q5) from a plane Γ external to the secant variety to Σ [G. Lunardon, O. Polverino: Translation ovoids of orthogonal polar spaces, Forum Math. 16 (2004) 663–669]. The pair (Γ, Σ) will be called a projecting configuration for the linear set. The projecting configurations for the only known maximum scattered linear sets in PG(1, q5), namely those of pseudoregulus and LP type, have been characterized in the literature [B. Csajbòk, C. Zanella: On scattered linear sets of pseudoregulus type in PG(1,qt) Finite Fields Appl. 41 (2016), 34–54; C. Zanella, F. Zullo: Vertex properties of maximum scattered linear sets of PG(1, qn). Discrete Math. 343 (2020) 111800]. Let (Γ, Σ) be a projecting configuration for a maximum scattered linear set in PG(1, q5). Let σ be a generator of G = PΓL(5, q^5)Σ, the pointwise stabilizer of Σ in PGL(5,q^5) and A = Γ ∩ Γσ^4, B = Γ ∩ Γσ^3. If A and B are not both points, then the projected linear set is of pseudoregulus type. Suppose that they are points. Recall that the rank of a point X is the vectorial dimension of the span of the orbit of X under the action of G. In this paper, by investigating the geometric properties of projecting configurations, it is proved that if at least one of the points A and B has rank 5, the associated maximum scattered linear set must be of LP type. Then, if a maximum scattered linear set of a new type exists, it must be such that rk A = rk B = 4. In this paper we derive two possible polynomial forms that such a linear set must have. An exhaustive analysis by computer shows that for q ≤ 25 no new maximum scattered linear set exists.
linear set, finite projective line, subgeometry, linearized polynomial, q-polynomial
Copyright (c) 2026 Algebraic Combinatorics
Descent set distribution for permutations with cycles of only odd or only even lengths
It is known that the number of permutations in the symmetric group $S_{2n}$ with cycles of odd lengths only is equal to the number of permutations with cycles of even lengths only. We prove a refinement of this equality, involving descent sets: the number of permutations in $S_{2n}$ with a prescribed descent set and all cycles of odd lengths is equal to the number of permutations with the complementary descent set and all cycles of even lengths. There is also a variant for $S_{2n+1}$. The proof uses generating functions for character values and applies a new identity on higher Lie characters.
Permutation, descent set, higher Lie character, character generating function
The geometric Tevelev degrees of projective space enumerate general, pointed algebraic curves interpolating through the maximal possible number of points. Previous work expresses these invariants in terms of Schubert calculus. Extending ideas of Gillespie--Reimer-Berg, we use the RSK correspondence to give a positive interpretation of these counts in terms of the combinatorics of words.
Restriction coefficients for partitions with at most three columns
Let $r \geq 0$, and let $\lambda$ and $\mu$ be partitions such that $\lambda_1 \leq r + 1$. We present a combinatorial interpretation of the plethysm coefficient $\langle s_\lambda, s_\mu[s_r] \rangle$. As a consequence, we solve the \emph{restriction problem} for partitions with at most three columns. That is, for all partitions $\lambda$ with $\lambda_1 \leq 3$, we find a combinatorial interpretation for the multiplicities of the irreducible $\mathfrak{S}_n$-submodules of the Schur module $\mathbb{S}^\lambda \mathbb{C}^n$, considered as an $\mathfrak{S}_n$-module.
Uniqueness for 2-Intersecting Families of Permutations and Perfect Matchings
We characterize the largest 2-intersecting families of permutations on n symbols and perfect matchings of the complete 2n-vertex graph for all n.
Representation Theory of the Symmetric Group
Chromatic symmetric functions and change of basis
We prove necessary conditions for certain elementary symmetric functions, $e_\la$, to appear with nonzero coefficient in Stanley's chromatic symmetric function as well as in the generalization considered by Shareshian and Wachs. We do this by first considering the expansion in the monomial or Schur basis and then performing a basis change. Using the former, we make a connection with two fundamental graph theory invariants, the independence and clique numbers. This allows us to prove nonnegativity of three-column coefficients for all natural unit interval graphs. The Schur basis permits us to give a new interpretation of the coefficient of $e_n$ in terms of tableaux. We are also able to give an explicit formula for that coefficient.
When is the Chromatic Quasisymmetric Function Symmetric?
We investigate the problem of when a chromatic quasisymmetric function (CQF) $X_G(x;q)$ of a graph $G$ is in fact symmetric. We first prove the remarkable fact that if a product of two quasisymmetric functions $f$ and $g$ in countably infinitely many variables is symmetric, then in fact $f$ and $g$ must be symmetric. This allows the problem to be reduced to the case of connected graphs. We then show that any labeled graph having more than one source or sink has a nonsymmetric CQF. As a corollary, we find that all trees other than a directed path have a nonsymmetric CQF. We also show that a family of graphs we call ``mixed mountain graphs'' always have symmetric CQF.
Copyright (c) 2025 Algebraic Combinatorics
We provide a formula for the Garsia-Remmel q-rook numbers as a sum over standard Young tableaux. We connect our formula with the coefficients in q- Whittaker expansion of unicellular LLT functions.
Geometries with trialities arising from linear spaces
A triality is a sort of super-symmetry that exchanges the types of the elements of an incidence geometry in cycles of length three. Although geometries with trialities exhibit fascinating behaviors, their construction is challenging, making them rare in the literature. To understand trialities more deeply, it is crucial to have a wide variety of examples at hand. In this article, we introduce a general method for constructing various rank-three incidence systems with trialities. Specifically, for any rank two incidence system $\Gamma$, we define its triangle complex $\Delta(\Gamma)$, a rank three incidence system whose elements consist of three copies of the flags (pairs of incident elements) of $\Gamma$. This triangle complex always admits a triality that cyclically permutes the three copies. We then explore in detail the properties of the triangle complex when $\Gamma$ is a linear space, including flag-transitivity, the existence of dualities, and connectivity properties. As a consequence of our work, this construction yields the first infinite family of thick, flag-transitive and residually connected geometries with trialities but no dualities.
Boundedness criteria for real quivers of rank 3
We study the boundedness of a mutation class for quivers with real weights. The main result is a characterization of bounded mutation classes for real quivers of rank 3.
Characterizing positroid quotients of uniform matroids
We study two-step flag positroids $(P_1, P_2)$, where $P_1$ is a quotient of $P_{2}$. We provide a complete characterization of all two-step flag positroids that contain a uniform matroid, extending and completing a partial result by Benedetti, Ch\'avez, and Jim\'enez. Our characterization reveals that ``"non-local information'' is necessary for characterizing flag positroids. To contrast general positroids with the special case of lattice path matroids, we show that the containment relations of Grassmann necklaces and conecklaces fully characterize flag lattice path matroids, but are insufficient for general flag positroids. Additionally, we prove that the decorated permutations of any elementary quotient pair are related by a cyclic shift, resolving a conjecture of Benedetti, Ch\'avez and Jim\'enez.
Upho lattices II: ways of realizing a core
A poset is called upper homogeneous, or “upho,” if all of its principal order filters are isomorphic to the whole poset. In previous work of the first author, it was shown that each (finite-type $\mathbb{N}$-graded) upho lattice has associated to it a finite graded lattice, called its core, which determines the rank generating function of the upho lattice. In that prior work the question of which finite graded lattices arise as cores was explored. Here, we study the question of in how many different ways a given finite graded lattice can be realized as the core of an upho lattice. We show that if the finite lattice has no nontrivial automorphisms, then it is the core of finitely many upho lattices. We also show that the number of ways a finite lattice can be realized as a core is unbounded, even when restricting to rank-two lattices. We end with a discussion of a potential algorithm for listing all the ways to realize a given finite lattice as a core.
A quasigroup is a pair $(Q, \cdot)$ where $Q$ is a non-empty set and $\cdot$ is a binary operation on $Q$ such that for every $(u, v) \in Q^2$ there exists a unique $(x, y) \in Q^2$ such that $u \cdot x = v = y \cdot u$. Let $q$ be an odd prime power, let $\mathbb{F}_q$ denote the finite field of order $q$, and let $\mathcal{R}_q$ denote the set of non-zero squares in $\mathbb{F}_q$. Let $\{a, b\} \subseteq \mathbb{F}_q$ be such that $\{ab, (a-1)(b-1)\} \subseteq \mathcal{R}_q$. Let $\mathcal{Q}_{a, b}$ denote the quadratic quasigroup $(\mathbb{F}_q, *_{a, b})$ where $*_{a, b}$ is defined by \[ x*_{a, b}y = \begin{cases} x+a(y-x) & \text{if } y-x \in \mathcal{R}_q, \\ x+b(y-x) & \text{otherwise}. \end{cases} \] The operation table of a quadratic quasigroup is a quadratic Latin square. Recently, it has been determined exactly when two quadratic quasigroups are isomorphic and the automorphism group of any quadratic quasigroup has been determined. In this paper, we extend these results. We determine exactly when two quadratic quasigroups are isotopic and we determine the autotopism group of any quadratic quasigroup. In the process, we count the number of $2 \times 2$ subsquares in quadratic Latin squares.
In a 2018 paper, Davis and Sagan studied several pattern-avoiding polytopes. They found that a particular pattern-avoiding Birkhoff polytope had the same normalized volume as the order polytope of a certain poset, leading them to ask if the two polytopes were unimodularly equivalent. Motivated by Davis and Sagan's question, in this paper we define a pattern-avoiding Birkhoff polytope called a $c$-Birkhoff polytope for each Coxeter element $c$ of the symmetric group. We then show that the $c$-Birkhoff polytope is unimodularly equivalent to the order polytope of the heap poset of the $c$-sorting word of the longest permutation. When $c=s_1s_2\dots s_{n}$, this result recovers an affirmative answer to Davis and Sagan's question. Another consequence of this result is that the normalized volume of the $c$-Birkhoff polytope is the number of the longest chains in the (type A) $c$-Cambrian lattice.
Decomposition numbers for weight 3 blocks of Iwahori–Hecke algebras of type B
Let B be a weight-3 block of an Iwahori–Hecke algebra of type B over any field. We develop the combinatorics of B to prove that the decomposition numbers for B are all 0 or 1.
Iwahori-Hecke algebra; Specht module; decomposition number; bipartition.
Lefschetz properties of local face modules
Local faces modules are modules over face rings whose Hilbert function is the local h-vector of a triangulation of a simplex. We study when Lefschetz properties hold for local face modules. We prove new unimodality results for local h-vectors of vertex-induced triangulations and show that, even for regular triangulations, Lefschetz properties can fail for local face modules in positive characteristic.
Weingarten calculus for centered random permutation matrices
We introduce and study the Weingarten calculus for centered random permutation matrices in the symmetric group $S_N$. After presenting a formulation of the Weingarten calculus on the symmetric group, we derive a formula in the centered case, as well as a sign-respecting formula. Our investigations uncover the fact that a building block of this Weingarten calculus is Kummer's confluent hypergeometric function. It allows us to derive multiple algebraic properties of the Weingarten function and uniform estimate. These results shed a conceptual light on phenomena that take place regarding the algebraic and asymptotic behavior of moments of random permutations in the resolution of Bordenave and Bordenave-Collins of strong convergence. We obtain multiple new non-trivial estimates for moments of coefficients in centered moments.
Eigenvalues of Heckman-Polychronakos operators
Heckman-Polychronakos operators form a prominent family of commuting differential--difference operators defined in terms of the Dunkl operators $\mathcal D_i$ as $\mathcal P_m= \sum_{i=1}^N (x_i \mathcal D_i)^m$. They have been known since 1990s in connection with trigonometric Calogero-Moser-Sutherland Hamiltonian and Jack symmetric polynomials. We explicitly compute the eigenvalues of these operators for symmetric and skew-symmetric eigenfunctions, as well as partial sums of eigenvalues for general polynomial eigenfunctions.
We prove that the existence of a divisor of degree 3 and Baker-Norine rank at least 1 on a 3-edge connected tropical curve is equivalent to the existence of a non-degenerate harmonic morphism of degree 3 from a tropical modification of it to a tropical rational curve. Using the second description, we define the moduli spaces of 3-edge connected tropical trigonal covers and of 3-edge connected tropical trigonal curves, the latter as a locus in the moduli space of tropical curves. Finally, we prove that the moduli space of 3-edge connected genus g tropical trigonal curves has the same dimension as the moduli space of genus g algebraic trigonal curves.
Snake graphs are a class of planar graphs that are important in the theory of cluster algebras. Indeed, the Laurent expansions of the cluster variables in cluster algebras from surfaces are given as weight generating functions for 1-dimer covers (or perfect matchings) of snake graphs. Moreover, the enumeration of 1-dimer covers of snake graphs provides a combinatorial interpretation of continued fractions. In particular, the number of 1-dimer covers of the snake graph $\mathcal{G}[a_1,\dots,a_n]$ is the numerator of the continued fraction $[a_1,\dots,a_n]$. This number is equal to the top left entry of the matrix product $\begin{pmatrix}a_1 & 1 \\ 1 & 0 \end{pmatrix} \cdots \begin{pmatrix} a_n & 1 \\ 1 & 0 \end{pmatrix}$. In this paper, we give enumerative results on $m$-dimer covers of snake graphs. We show that the number of $m$-dimer covers of the snake graph $\mathcal{G}[a_1,\dots,a_n]$ is the top left entry of a product of analogous $(m+1)$-by-$(m+1)$ matrices. We discuss how our enumerative results are related to other known combinatorial formulas, and we suggest a generalization of continued fractions based on our methods. These generalized continued fractions provide some interesting open questions and a possibly novel approach towards Hermite's problem for cubic irrationals.
Perfect state transfer in graphs related to linear groups in two dimensions
We construct families of graphs from linear groups SL(2,q), GL(2,q) and GU(2,q^2), where $q$ is an odd prime power, with the property that the continuous-time quantum walks on the associated networks of qubits admit perfect state transfer.
Codegree and regularity of stable set polytopes
The codegree ${\rm codeg}(P)$ of a lattice polytope $P$ is a fundamental invariant in discrete geometry. In the present paper, we investigate the codegree of the stable set polytope $P_G$ associated with a graph $G$. Specifically, we establish the inequalities \[ \omega(G) + 1 \leq {\rm codeg}(P_G) \leq \chi(G) + 1, \] where $\omega(G)$ and $\chi(G)$ denote the clique number and the chromatic number of G, respectively. Furthermore, an explicit formula for ${\rm codeg}(P_G$ is given when $G$ is either a line graph or an $h$-perfect graph. Finally, as an application of these results, we provide upper and lower bounds on the regularity of the toric ring associated with $P_G$.
stable set polytope, codegree, clique number, chromatic number, matching polytope, perfect graph, $h$-perfect graph, toric ring, regularity
More on the corner-vector construction for spherical designs
This paper explores a full generalization of the classical corner-vector method for constructing weighted spherical designs, which we call the generalized corner-vector method. First we establish a uniform upper bound for the degree of designs obtained from the proposed method. Our proof is a hybrid argument that employs techniques in analysis and combinatorics, especially a famous result by Xu(1998) on the interrelation between spherical designs and simplical designs, and the cross-ratio comparison method for Hilbert identities introduced by Nozaki and Sawa(2013). We extensively study conditions for the existence of designs obtained from our method, and present many curious examples of degree 7 through 13, some of which are, to our surprise, characterized in terms of integral lattices.
Real and Positive Tropicalizations of Symmetric Determinantal Varieties
We study real and positive tropicalizations of the varieties of low rank symmetric matrices over real or complex Puiseux series. We show that real tropicalization coincides with complex tropicalization for rank two and corank one cases. We also show that the two notions of positive tropicalization introduced by Speyer and Williams coincide for symmetric rank two matrices, but they differ for symmetric corank one matrices.
Numerical semigroups, polyhedra, and posets IV: walking the faces of the Kunz cone
A numerical semigroup is a cofinite subset of $\mathbb Z_{\ge 0}$ containing $0$ and closed under addition. Each numerical semigroup $S$ with smallest positive element $m$ corresponds to an integer point in the Kunz cone $\mathcal C_m \subseteq \mathbb R^{m-1}$, and the face of $\mathcal C_m$ containing that integer point determines certain algebraic properties of $S$. In this paper, we introduce the Kunz fan, a pure, polyhedral cone complex comprised of a faithful projection of certain faces of $\mathcal C_m$. We characterize several aspects of the Kunz fan in terms of the combinatorics of Kunz nilsemigroups, which are known to index the faces of $\mathcal C_m$, and our results culminate in a method of ``walking'' the face lattice of the Kunz cone in a manner analogous to that of a Gr\"obner walk. We apply our results in several contexts, including a wealth of computational data obtained from the aforementioned ``walks'' and a proof of a recent conjecture concerning which numerical semigroups achieve the highest minimal presentation cardinality when one fixes the smallest positive element and the number of generators.
A conjectural basis for the $(1,2)$-bosonic-fermionic coinvariant ring
We give the first conjectural construction of a monomial basis for the coinvariant ring $R_n^{(1,2)}$, for the symmetric group $\mathfrak{S}_n$ acting on one set of bosonic (commuting) and two sets of fermionic (anticommuting) variables. Our construction interpolates between the modified Motzkin path basis for $R_n^{(0,2)}$ of Kim--Rhoades (2022) and the super-Artin basis for $R_n^{(1,1)}$ conjectured by Sagan--Swanson (2024) and proven by Angarone et al. (2024). We prove that our proposed basis has cardinality $2^{n-1}n!$, aligning with a conjecture of Zabrocki (2020) on the dimension of $R_n^{(1,2)}$, and show how it gives a combinatorial expression for the Hilbert series. We also conjecture a Frobenius series for $R_n^{(1,2)}$. We show that these proposed Hilbert and Frobenius series are equivalent to conjectures of Iraci, Nadeau, and Vanden Wyngaerd (2023) on $R_n^{(1,2)}$ in terms of segmented Smirnov words, by exhibiting a weight-preserving bijection between our proposed basis and their segmented permutations. We extend some of their results on the sign character to hook characters, and give a formula for the $m_\mu$ coefficients of the conjectural Frobenius series. Finally, we conjecture a monomial basis for the analogous ring in type $B_n$, and show that it has cardinality $4^nn!$.
A type $B$ analog of the Ish arrangement
The Shi arrangement due to Shi (1986) and the Ish arrangement due to Armstrong (2013) are deformations of the type $A$ Coxeter arrangement that share many common properties. Motivated by a question of Armstrong and Rhoades since 2012 to seek for Ish arrangements of other types, in this paper we introduce an Ish arrangement of type $B$. We study this Ish arrangement through various aspects similar to as known in type $A$ with a main emphasis on freeness and supersolvability. Our method is based on the concept of $\psi$-digraphic arrangements recently introduced due to Abe and the authors with a type $B$ extension.
Web invariants for flamingo Specht modules
Webs yield an especially important realization of certain Specht modules, irreducible representations of symmetric groups, as they provide a pictorial basis with a convenient diagrammatic calculus. In recent work, the last three authors associated polynomials to noncrossing partitions without singleton blocks, so that the corresponding polynomials form a web basis of the pennant Specht module $S^{(d,d,1^{n-2d})}$. These polynomials were interpreted as global sections of a line bundle on a $2$-step partial flag variety. Here, we both simplify and extend this construction. On the one hand, we show that these polynomials can alternatively be situated in the homogeneous coordinate ring of a Grassmannian, instead of a $2$-step partial flag variety, and can be realized as tensor invariants of classical (but highly nonplanar) tensor diagrams. On the other hand, we extend these ideas from the pennant Specht module $S^{(d,d,1^{n-2d})}$ to more general flamingo Specht modules $S^{(d^r,1^{n-rd})}$. In the hook case $r=1$, we obtain a spanning set that can be restricted to a basis in various ways. In the case $r>2$, we obtain a basis of a well-behaved subspace of $S^{(d^r,1^{n-rd})}$, but not of the entire module.
Tridendriform algebras on hypergraph polytopes
We extend the works of Loday-Ronco and Burgunder-Ronco on the tridendriform decomposition of the shuffle product on the faces of associahedra and permutohedra, to other families of hypergraph polytopes (or nestohedra), including simplices, hypercubes and some new families. We also extend the shuffle product to take more than two arguments, and define accordingly a new algebraic structure, that we call polydendriform, from which the original tridendriform equations can be crisply synthesized.
HIVE-TYPE POLYTOPES FOR QUIVER SEMI-INVARIANTS AND THE MEMBERSHIP PROBLEM FOR QUIVER MOMENT CONES
Let $\mathcal{Q}$ be a bipartite quiver with vertex set $\mathcal{Q}_0$ such that the number of arrows between any source vertex and any sink vertex is constant. Let $\beta=(\beta(x))_{x \in \mathcal{Q}_0}$ be a dimension vector of $\mathcal{Q}$ with positive integer coordinates. Let $\mathcal{Q}_{\beta}$ be the flag-extension of $\mathcal{Q}$ obtained by attaching a flag $\mathcal{F}(x)$ of length $\beta(x)-1$ at every vertex $x$ of $\mathcal{Q}$, and let $\widetilde{\beta}$ be the extension of $\beta$ to $\mathcal{Q}_{\beta}$ that takes values $1, \ldots, \beta(x)$ along the vertices of the flag $\F(x)$ for every vertex $x$ of $\mathcal{Q}$. For an integral weight $\widetilde{\sigma}$ of $\mathcal{Q}_{\beta}$, let $K_{\widetilde{\sigma}}$ be the dimension of the space of semi-invariants of weight $\widetilde{\sigma}$ on the representation space of $\widetilde{\beta}$-dimensional complex representations of $\mathcal{Q}_{\beta}$. We show that $K_{\widetilde{\sigma}}$ can be expressed as the number of lattice points of a polytope obtained by gluing together two Knutson-Tao hive polytopes. Furthermore, this polytopal description together with Derksen-Weyman's Saturation Theorem for quiver semi-invariants allows us to use Tardos' algorithm to solve the membership problem for the moment cone associated to $(\mathcal{Q},\beta)$ in strongly polynomial time.
Hive polytopes, Littlewood-Richardson coefficients, moment cones, strongly polynomial time, quiver exceptional sequences, quiver semi-invariants.