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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
Copyright (c) 2026 Algebraic Combinatorics
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.
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.
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.
Projective dimension of weakly chordal graphic arrangements
A graphic arrangement is a subarrangement of the braid arrangement whose set of hyperplanes is determined by an undirected graph. A classical result due to Stanley, Edelman and Reiner states that a graphic arrangement is free if and only if the corresponding graph is chordal, i.e., the graph has no chordless cycle with four or more vertices. In this article we extend this result by proving that the module of logarithmic derivations of a graphic arrangement has projective dimension at most one if and only if the corresponding graph is weakly chordal, i.e., the graph and its complement have no chordless cycle with five or more vertices.
Crystals for set-valued decomposition tableaux
We describe two crystal structures on set-valued decomposition tableaux. These provide the first examples of interesting ``$K$-theoretic'' crystals on shifted tableaux. Our first crystal is modeled on a similar construction of Monical, Pechenik, and Scrimshaw for semistandard (unshifted) set-valued tableaux. Our second crystal is adapted from the ``square root'' operators introduced by Yu on the same set. Neither of our shifted crystals is normal, but we conjecture that our second construction is connected with a unique highest weight element. These results lead to partial progress on a conjectural formula of Cho--Ikeda for $K$-theoretic Schur $P$-functions. We also study a new category of ``square root crystals'' that includes our second construction and Yu's set-valued tableau crystals as examples. We observe that Buch's formula for the coefficients expanding products of symmetric Grothendieck functions has a simple description in terms of the tensor product for this category.
We show that for any permutation w, the increasing flip graph on acyclic pipe dreams with exiting permutation w is a lattice quotient of the interval [e,w] of the weak order. We then discuss conjectural generalizations of this result to acyclic facets of subword complexes on arbitrary finite Coxeter groups.
Centraliser algebras of monomial representations and applications in combinatorics
Centraliser algebras of monomial representations of finite groups may be constructed and studied using methods similar to those employed in the study of permutation groups. Guided by results of D. G. Higman and others, we give an explicit construction for a basis of the centraliser algebra of a monomial representation. The character table of this algebra is then constructed via character sums over double cosets. We locate the theory of group-developed and cocyclic-developed Hadamard matrices within this framework. We apply Grobner bases to produce a new classification of highly symmetric complex Hadamard matrices. MSC: 05B20, 20B25.
Tropical Fock-Goncharov coordinates \\ for $\operatorname{SL}(3)$-webs on surfaces II: naturality
In \cite{DouglasArxiv20}, we constructed nonnegative integer coordinates $\Phi_\mathscr{T}(\mathscr{W}_{3, \widehat{S}}) \subset \mathbb{Z}_{\geq 0}^N$ for the collection $\mathscr{W}_{3, \widehat{S}}$ of reduced $\mathrm{SL}_3$-webs on a finite-type punctured surface $\widehat{S}$, depending on an ideal triangulation $\mathscr{T}$ of $\widehat{S}$. We show that these coordinates are natural with respect to the choice of triangulation, in the sense that if a different triangulation $\mathscr{T}^\prime$ is chosen, then the coordinate change map relating $\Phi_\mathscr{T}(\mathscr{W}_{3, \widehat{S}})$ to $\Phi_{\mathscr{T}^\prime}(\mathscr{W}_{3, \widehat{S}})$ is a tropical $\mathcal{A}$-coordinate cluster transformation. We can therefore view the webs $\mathscr{W}_{3, \widehat{S}}$ as a concrete topological model for the Fock-Goncharov-Shen positive integer tropical points~$\mathcal{A}_{\mathrm{PGL}_3, \widehat{S}}^+(\mathbb{Z}^t)$.
Root polytopes, tropical types, and toric edge ideals
We consider arrangements of tropical hyperplanes where the apices of the hyperplanes are taken to infinity in certain directions. Such an arrangement defines a decomposition of Euclidean space where a cell is determined by its `type' data, analogous to the covectors of an oriented matroid. By work of Develin-Sturmfels and Fink-Rincon, these `tropical complexes' are dual to (regular) subdivisions of root polytopes, which in turn are in bijection with mixed subdivisions of certain generalized permutohedra. Extending previous work with Joswig-Sanyal, we show how a natural monomial labeling of these complexes describes polynomial relations (syzygies) among `type ideals' which arise naturally from the combinatorial data of the arrangement. In particular, we show that the cotype ideal is Alexander dual to a corresponding initial ideal of the lattice ideal of the underlying root polytope. This leads to novel ways of studying algebraic properties of various monomial and toric ideals, as well as relating them to combinatorial and geometric properties. In particular, our methods of studying the dimension of the tropical complex leads to new formulas for homological invariants of toric edge ideals of bipartite graphs, which have been extensively studied in the commutative algebra community.
We investigate the ``stratified Ehrhart ring theory'' for periodic graphs, which gives an algorithm for determining the growth sequences of periodic graphs. The growth sequence $(s_{\Gamma, x_0, i})_{i \ge 0}$ is defined for a graph $\Gamma$ and its fixed vertex $x_0$, where $s_{\Gamma, x_0, i}$ is defined as the number of vertices of $\Gamma$ at distance $i$ from $x_0$. Although the sequences $(s_{\Gamma, x_0, i})_{i \ge 0}$ for periodic graphs are known to be of quasi-polynomial type, their determination had not been established, even in dimension two. Our theory and algorithm can be applied to arbitrary periodic graphs of any dimension. As an application of the algorithm, we determine the growth sequences in several new examples.
periodic graphs, growth sequence, growth series, Ehrhart theory, convex geometry
We prove that the tropical Abel--Prym map $\Psi\colon \tGa\to\Prym(\tGa/\Ga)$ associated with a free double cover $\pi\colon \tGa\to \Ga$ of hyperelliptic metric graphs is harmonic of degree $2$ in accordance with the already established algebraic result. We then prove a partial converse. Contrary to the analogous algebraic result, when the source graph of the double cover is not hyperelliptic, the Abel--Prym map is often not injective. When the source graph is hyperelliptic, we show that the Abel--Prym graph $\Psi(\tGa)$ is a hyperelliptic metric graph of genus $g_{\Ga}-1$ whose Jacobian is isomorphic, as pptav, to the Prym variety of the cover. En route, we count the number of distinct free double covers by hyperelliptic metric graphs.
Integer Diagonal Forms for Subset Intersection Relations
For integers $0 \leq \ell \leq k_{r} \leq k_{c} \leq n$, we give a description for the Smith group of the incidence matrix with rows (columns) indexed by the size $k_r$ ($k_c$, respectively) subsets of an $n$-element set, where incidence means intersection in a set of size $\ell$. This generalizes work of Wilson and Bier from the 1990s which dealt only with the case where incidence meant inclusion. Our approach also describes the Smith group of any matrix in the $\Z$-linear span of these matrices so includes all integer matrices in the Bose-Mesner algebra of the Johnson association scheme: for example, the association matrices themselves as well as the Laplacian, signless Laplacian, Seidel adjacency matrix, etc. of the associated graphs. In particular, we describe the critical (also known as sandpile) groups of these graphs. The complexity of our formula grows with the $k$ parameters, but is independent of $n$ and $\ell$, which often leads to an efficient algorithm for computing these groups. We illustrate our techniques to give diagonal forms of matrices attached to the Kneser and Johnson graphs for subsets of size $3$, whose invariants have never before been described, and recover results from a variety of papers in the literature in a unified way.