Combinatorial Formula Research Articles

A matrix theory introduction to seaweed algebras and their index

The index of a Lie algebra is an important algebraic invariant, but it is notoriously difficult to compute. However, for the suggestively-named seaweed algebras, the computation of the index can be reduced to a combinatorial formula based on the connected components of a “meander”: a planar graph associated with the algebra. Our index analysis on seaweed algebras requires only basic linear and abstract algebra. Indeed, the main goal of this article is to introduce a broader audience to seaweed algebras with minimal appeal to specialized language and notation from Lie theory. This said, we present several results that do not appear elsewhere and do appeal to more advanced language in the Introduction to provide added context.

Fertilitopes

We introduce tools from discrete convexity theory and polyhedral geometry into the theory of West’s stack-sorting map s. Associated to each permutation pi is a particular set mathcal V(pi ) of integer compositions that appears in a formula for the fertility of pi , which is defined to be |s^{-1}(pi )|. These compositions also feature prominently in more general formulas involving families of colored binary plane trees called troupes and in a formula that converts from free to classical cumulants in noncommutative probability theory. We show that mathcal V(pi ) is a transversal discrete polymatroid when it is nonempty. We define the fertilitope of pi to be the convex hull of mathcal V(pi ), and we prove a surprisingly simple characterization of fertilitopes as nestohedra arising from full binary plane trees. Using known facts about nestohedra, we provide a procedure for describing the structure of the fertilitope of pi directly from pi using Bousquet-Mélou’s notion of the canonical tree of pi . As a byproduct, we obtain a new combinatorial cumulant conversion formula in terms of generalizations of canonical trees that we call quasicanonical trees. We also apply our results on fertilitopes to study combinatorial properties of the stack-sorting map. In particular, we show that the set of fertility numbers has density 1, and we determine all infertility numbers of size at most 126. Finally, we reformulate the conjecture that sum _{sigma in s^{-1}(pi )}x^{textrm{des}(sigma )+1} is always real-rooted in terms of nestohedra, and we propose natural ways in which this new version of the conjecture could be extended.

Combinatorial Sums and Identities associated with Functional Equations of Generating Functions for Fubini Type Polynomials

Using generating functions with their functional equations method, a great number of novel combinatorial sums, formulas, and recurrence relation including Fubini type polynomials and numbers, Stirling type numbers, and Apostol type polynomials are given. Applying Riemann integral to this generating function with their functional equations, some identities involving Cauchy and Stirling numbers are obtained. Moreover, some interpretations about the results are given.

Mixed Dimer Configuration Model in Type D Cluster Algebras

We define a combinatorial model for $F$-polynomials and $g$-vectors for type $D_n$ cluster algebras where the associated quiver is acyclic. Our model utilizes a combination of dimer configurations and double dimer configurations which we refer to as mixed dimer configurations. In particular, we give a graph theoretic recipe that describes which monomials appear in such $F$-polynomials, as well as a graph theoretic way to determine the coefficients of each of these monomials. In addition, we prove that a weighting on our mixed dimer configuration model yields the associated $g$-vector. To prove this formula, we use a combinatorial formula due to Thao Tran (arXiv:0911.4462, 2009) and provide explicit bijections between her combinatorial model and our own.

Inverse K-Chevalley formulas for semi-infinite flag manifolds, II: Arbitrary weights in ADE type

We continue the study, begun in [11], of inverse Chevalley formulas for the equivariant K-group of semi-infinite flag manifolds. Using the language of alcove paths, we reformulate and extend our combinatorial inverse Chevalley formula to arbitrary weights in all simply-laced types (conjecturally also in type E8).

Palindromic and Colored Superdiagonal Compositions

A superdiagonal composition is one in which the i-th part or summand is of size greater than or equal to i. In this paper, we study the number of palindromic superdiagonal compositions and colored superdiagonal compositions. In particular, we give generating functions and explicit combinatorial formulas involving binomial coefficients and Stirling numbers of the first kind.

Grothendieck-to-Lascoux expansions

We establish the conjecture of Reiner and Yong for an explicit combinatorial formula for the expansion of a Grothendieck polynomial into the basis of Lascoux polynomials. This expansion is a subtle refinement of its symmetric function version due to Buch, Kresch, Shimozono, Tamvakis, and Yong, which gives the expansion of stable Grothendieck polynomials indexed by permutations into Grassmannian stable Grothendieck polynomials. Our expansion is the K K -theoretic analogue of that of a Schubert polynomial into Demazure characters, whose symmetric analogue is the expansion of a Stanley symmetric function into Schur functions. Our expansions extend to flagged Grothendieck polynomials.

Matrix formulae for decorated super Teichmüller spaces

For an arc on a bordered surface with marked points, we associate a holonomy matrix using a product of elements of the supergroup OSp(1|2), which defines a flat OSp(1|2)-connection on the surface. We show that our matrix formula of an arc yields its super λ-length in Penner-Zeitlin's decorated super Teichmüller space. This generalizes the matrix formula of Fock-Goncharov and Musiker-Williams. We also prove that our matrix formulas agree with the combinatorial formulas given in the authors' previous works. As an application, we use our matrix formula in the case of an annulus to obtain new results on super Fibonacci numbers.

Gruson–Serganova character formulas and the Duflo–Serganova cohomology functor

Abstract We establish an explicit formula for the character of an irreducible finite-dimensional representation of g ⁢ l ⁢ ( m | n ) \mathfrak{gl}(m|n) . The formula is a finite sum with integer coefficients in terms of a basis E μ \mathcal{E}_{\mu} (Euler characters) of the character ring. We prove a simple formula for the behavior of the “superversion” of E μ \mathcal{E}_{\mu} in the g ⁢ l ⁢ ( m | n ) \mathfrak{gl}(m|n) and o ⁢ s ⁢ p ⁢ ( m | 2 ⁢ n ) \mathfrak{osp}(m|2n) -case under the map ds \mathrm{ds} on the supercharacter ring induced by the Duflo–Serganova cohomology functor DS \mathrm{DS} . As an application, we get combinatorial formulas for superdimensions, dimensions and g 0 \mathfrak{g}_{0} -decompositions for g ⁢ l ⁢ ( m | n ) \mathfrak{gl}(m|n) and o ⁢ s ⁢ p ⁢ ( m | 2 ⁢ n ) \mathfrak{osp}(m|2n) .

Finite free cumulants: Multiplicative convolutions, genus expansion and infinitesimal distributions

Given two polynomials p ( x ) , q ( x ) p(x), q(x) of degree d d , we give a combinatorial formula for the finite free cumulants of p ( x ) ⊠ d q ( x ) p(x)\boxtimes _d q(x) . We show that this formula admits a topological expansion in terms of non-crossing multi-annular permutations on surfaces of different genera. This topological expansion, on the one hand, deepens the connection between the theories of finite free probability and free probability, and in particular proves that ⊠ d \boxtimes _d converges to ⊠ \boxtimes as d d goes to infinity. On the other hand, borrowing tools from the theory of second order freeness, we use our expansion to study the infinitesimal distribution of certain families of polynomials which include Hermite and Laguerre, and draw some connections with the theory of infinitesimal distributions for real random matrices. Finally, building on our results we give a new short and conceptual proof of a recent result (see J. Hoskins and Z. Kabluchko [Exp. Math. (2021), pp. 1–27]; S. Steinerberger [Exp. Math. (2021), pp. 1–6]) that connects root distributions of polynomial derivatives with free fractional convolution powers.

Set-valued tableaux rule for Lascoux polynomials

Lascoux polynomials generalize Grassmannian stable Grothendieck polynomials and may be viewed as K-theoretic analogs of key polynomials. The latter two polynomials have combinatorial formulas involving tableaux: Lascoux and Schützenberger gave a combinatorial formula for key polynomials using right keys; Buch gave a set-valued tableau formula for Grassmannian stable Grothendieck polynomials. We establish a novel combinatorial description for Lascoux polynomials involving right keys and set-valued tableaux. Our description generalizes the tableaux formulas of key polynomials and Grassmannian stable Grothendieck polynomials. To prove our description, we construct a new abstract Kashiwara crystal structure on set-valued tableaux. This construction answers an open problem of Monical, Pechenik and Scrimshaw.Mathematics Subject Classifications: 05E05Keywords: Lascoux polynomials, set-valued tableaux, crystal operators

Motivic zeta functions of hyperplane arrangements

AbstractFor each central essential hyperplane arrangement $\mathcal{A}$ over an algebraically closed field, let $Z_\mathcal{A}^{\hat\mu}(T)$ denote the Denef–Loeser motivic zeta function of $\mathcal{A}$ . We prove a formula expressing $Z_\mathcal{A}^{\hat\mu}(T)$ in terms of the Milnor fibers of related hyperplane arrangements. This formula shows that, in a precise sense, the degree to which $Z_{\mathcal{A}}^{\hat\mu}(T)$ fails to be a combinatorial invariant is completely controlled by these Milnor fibers. As one application, we use this formula to show that the map taking each complex arrangement $\mathcal{A}$ to the Hodge–Deligne specialization of $Z_{\mathcal{A}}^{\hat\mu}(T)$ is locally constant on the realization space of any loop-free matroid. We also prove a combinatorial formula expressing the motivic Igusa zeta function of $\mathcal{A}$ in terms of the characteristic polynomials of related arrangements.

Categories for Grassmannian Cluster Algebras of Infinite Rank

Abstract We construct Grassmannian categories of infinite rank, providing an infinite analogue of the Grassmannian cluster categories introduced by Jensen, King, and Su. Each Grassmannian category of infinite rank is given as the category of graded maximal Cohen–Macaulay modules over a certain hypersurface singularity. We show that generically free modules of rank $1$ in a Grassmannian category of infinite rank are in bijection with the Plücker coordinates in an appropriate Grassmannian cluster algebra of infinite rank. Moreover, this bijection is structure preserving, as it relates rigidity in the category to compatibility of Plücker coordinates. Along the way, we develop a combinatorial formula to compute the dimension of the $\textrm {Ext}^{1}$-spaces between any two generically free modules of rank $1$ in the Grassmannian category of infinite rank.

Motion planning in polyhedral products of groups and a Fadell-Husseini approach to topological complexity

We compute the topological complexity of a polyhedral product $\mathcal{Z}$ defined {in terms of} an $\operatorname{LS}$-logarithmic family of locally compact connected $\operatorname{CW}$ topological groups. The answer is given by a combinatorial formula that involves the $\operatorname{LS}$ category of the polyhedral-product factors. As a by-product, we show that the Iwase-Sakai conjecture holds true for $\mathcal{Z}$. The proof methodology {uses} a Fadell-Husseini viewpoint for the monoidal topological complexity $\big(\mathsf{TC}^M\big)$ of a space, which, under mild conditions, recovers Iwase-Sakai's original definition. In the Fadell-Husseini context, the stasis condition - $\mathsf{TC}^M$'s \emph{raison d'\^etre} - can be encoded at the covering level. Our Fadell-Husseini inspired definition provides an alternative to the $\mathsf{TC}^M$ variant given by Dranishnikov, as well as to the ones provided by Garc\'ia-Calcines, Carrasquel-Vera and Vandembroucq in terms of relative category.

Modified Macdonald polynomials and the multispecies zero-range process: I

In this paper we prove a new combinatorial formula for the modified Macdonald polynomials H ˜ λ (X;q,t), motivated by connections to the theory of interacting particle systems from statistical mechanics. The formula involves a new statistic called queue inversions on fillings of tableaux. This statistic is closely related to the multiline queues which were recently used to give a formula for the Macdonald polynomials P λ (X;q,t). In the case q=1 and X=(1,1,⋯,1), that formula had also been shown to compute stationary probabilities for a particle system known as the multispecies ASEP on a ring, and it is natural to ask whether a similar connection exists between the modified Macdonald polynomials and a suitable statistical mechanics model. In a sequel to this work, we demonstrate such a connection, showing that the stationary probabilities of the multispecies totally asymmetric zero-range process (mTAZRP) on a ring can be computed using tableaux formulas with the queue inversion statistic. This connection extends to arbitrary X=(x 1 ,⋯,x n ); the x i play the role of site-dependent jump rates for the mTAZRP.

A Tableau Formula for Vexillary Schubert Polynomials in Type C

Ikeda-Mihalcea-Naruse's double Schubert polynomials [Adv. Math. 226 (2011)] represent the equivariant cohomology classes of Schubert varieties in the type C flag varieties. The goal of this paper is to obtain a new tableau formula of these polynomials associated to vexillary signed permutations introduced by Anderson-Fulton. To achieve that goal, we introduce flagged factorial (Schur) $Q$-functions, combinatorially defined functions in terms of marked shifted tableaux for flagged strict partitions, and prove their Schur-Pfaffian formula. As an application, we also obtain a new combinatorial formula of factorial $Q$-functions of Ivanov in which monomials bijectively corresponds to flagged marked shifted tableaux.

Linear Transformations of Vertex Operators of Hall–Littlewood Polynomials

We study the effect of linear transformations on quantum fields with applications to vertex operator presentations of symmetric functions. Properties of linearly transformed quantum fields and corresponding transformations of Hall–Littlewood polynomials are described, including preservation of commutation relations, stability, explicit combinatorial formulas, and generating functions. We prove that specializations of linearly transformed Hall–Littlewood polynomials describe all polynomial tau-functions of the KP and BKP hierarchies. Examples of linear transformations are related to multiparameter symmetric functions, Grothendieck polynomials, deformations by cyclotomic polynomials, and some other variations of Schur symmetric functions that exist in the literature.

Matrix tree theorem for the net Laplacian matrix of a signed graph

For a simple signed graph G with the adjacency matrix A and net degree matrix , the net Laplacian matrix is . We introduce a new oriented incidence matrix which can keep track of the sign as well as the orientation of each edge of G. Also . Using this decomposition, we find the number of both positive and negative spanning trees of G in terms of the principal minors of generalizing the Matrix Tree Theorem for an unsigned graph. We present similar results for the signless net Laplacian matrix along with a combinatorial formula for its determinant.

Based modules over the $$\imath $$quantum group of type AI

This paper studies classical weight modules over the $$\imath $$ quantum group $$\textbf{U}^{\imath }$$ of type AI. We introduce the notion of based $$\textbf{U}^{\imath }$$ -modules by generalizing the notion of based modules over quantum groups (quantized enveloping algebras). We prove that each finite-dimensional irreducible classical weight $$\textbf{U}^{\imath }$$ -module with integer highest weight is a based $$\textbf{U}^{\imath }$$ -module. As a byproduct, a new combinatorial formula for the branching rule from $$\mathfrak {sl}_n$$ to $$\mathfrak {so}_n$$ is obtained.

An elementary approach to local combinatorial formulae for the Euler class of a PL spherical fibre bundle

We present an elementary approach to local combinatorial formulae for the Euler class of a fibre-oriented triangulated spherical fibre bundle. This approach is based on sections averaging technique and very basic knowledge of simplicial (co)homology theory. Our formulae are close relatives of those due to Mnëv. Bibliography: 9 titles.