• MacMahon Partition Analysis: A discrete approach to broken stick problems
J. Comb. Theory A (IF 1.192) Pub Date : 2021-11-30
William Verreault

We propose a discrete approach to solve problems on forming polygons from broken sticks, which is akin to counting polygons with sides of integer length subject to certain Diophantine inequalities. Namely, we use MacMahon's Partition Analysis to obtain a generating function for the size of the set of segments of a broken stick subject to these inequalities. In particular, we use this approach to show

• Symmetric binary Steinhaus triangles and parity-regular Steinhaus graphs
J. Comb. Theory A (IF 1.192) Pub Date : 2021-11-29
Jonathan Chappelon

A binary Steinhaus triangle is a triangle of zeroes and ones that points down and with the same local rule as the Pascal triangle modulo 2. A binary Steinhaus triangle is said to be rotationally symmetric, horizontally symmetric or dihedrally symmetric if it is invariant under the 120 degrees rotation, the horizontal reflection or both, respectively. The first part of this paper is devoted to the study

• On zero-sum subsequences of length kexp⁡(G) II
J. Comb. Theory A (IF 1.192) Pub Date : 2021-11-22
Weidong Gao, Siao Hong, Jiangtao Peng

Let G be an additive finite abelian group of exponent exp⁡(G). For every positive integer k, let skexp⁡(G)(G) denote the smallest integer t such that every sequence over G of length t has a zero-sum subsequence of length kexp⁡(G). Let ηkexp⁡(G)(G) denote the smallest integer t such that every sequence over G of length t has a zero-sum subsequence of length between 1 and kexp⁡(G). It is conjectured

• Infinite Sperner's theorem
J. Comb. Theory A (IF 1.192) Pub Date : 2021-11-17
Benny Sudakov, István Tomon, Adam Zsolt Wagner

One of the most classical results in extremal set theory is Sperner's theorem, which says that the largest antichain in the Boolean lattice 2[n] has size Θ(2nn). Motivated by an old problem of Erdős on the growth of infinite Sidon sequences, in this note we study the growth rate of maximum infinite antichains. Using the well known Kraft's inequality for prefix codes, it is not difficult to show that

• A new framework for identifying absolute maximum nonlinear functions
J. Comb. Theory A (IF 1.192) Pub Date : 2021-11-17
Bangteng Xu

Highly nonlinear functions (perfect nonlinear, maximum nonlinear, etc.) on finite fields and finite (abelian or nonabelian) groups have been studied in numerous papers. Among them are absolute maximum nonlinear functions on finite nonabelian groups introduced by Poinsot and Pott  in 2011. Recently, properties and constructions of absolute maximum nonlinear functions were studied in . In this

• Fixed points of group actions on link collapsible simplicial complexes
J. Comb. Theory A (IF 1.192) Pub Date : 2021-11-18
Michał J. Kukieła, Bernd S.W. Schröder

We prove that every action of a group on a link collapsible simplicial complex fixes a simplex. Under additional assumptions, the set of points fixed by the action induced on the geometric realization of the complex is contractible. In particular, it follows that the set of fixed points of the geometric realization of a simplicial endomorphism of a link collapsible clique complex is contractible and

• Combinatorics of quasi-hereditary structures
J. Comb. Theory A (IF 1.192) Pub Date : 2021-11-18
Manuel Flores, Yuta Kimura, Baptiste Rognerud

A quasi-hereditary algebra is an Artin algebra together with a partial order on its set of isomorphism classes of simple modules which satisfies certain conditions. In this article we investigate all the possible choices that yield quasi-hereditary structures on a given algebra, in particular we introduce and study what we call the poset of quasi-hereditary structures. Our techniques involve certain

• Independence of permutation limits at infinitely many scales
J. Comb. Theory A (IF 1.192) Pub Date : 2021-11-08
David Bevan

We introduce a new natural notion of convergence for permutations at any specified scale, in terms of the density of patterns of restricted width. In this setting we prove that limits may be chosen independently at a countably infinite number of scales.

• Combinatorics of continuants of continued fractions with 3 limits
J. Comb. Theory A (IF 1.192) Pub Date : 2021-10-29
Douglas Bowman, Herman D. Schaumburg

We give combinatorial descriptions of the terms occurring in continuants of general continued fractions that diverge to three limits. Equating this combinatorics with the usual combinatorial description due to Euler induces nontrivial identities. Special cases and applications to counting sequences are given.

• Confluence in labeled chip-firing
J. Comb. Theory A (IF 1.192) Pub Date : 2021-10-25
Caroline Klivans, Patrick Liscio

In 2016, Hopkins, McConville, and Propp proved that labeled chip-firing on a line always leaves the chips in sorted order provided that the initial number of chips is even. We present a novel proof of this result. We then apply our methods to resolve a number of related conjectures concerning the confluence of labeled chip-firing systems.

• Subdivisions of shellable complexes
J. Comb. Theory A (IF 1.192) Pub Date : 2021-10-22
Max Hlavacek, Liam Solus

In geometric, algebraic, and topological combinatorics, the unimodality of combinatorial generating polynomials is frequently studied. Unimodality follows when the polynomial is (real) stable, a property often deduced via the theory of interlacing polynomials. Many of the open questions on stability and unimodality of polynomials pertain to the enumeration of faces of cell complexes. In this paper

• A weak version of Kirillov's conjecture on Hecke–Grothendieck polynomials
J. Comb. Theory A (IF 1.192) Pub Date : 2021-10-21
Zerui Zhang, Yuqun Chen

Hecke-Grothendieck polynomials were introduced by Kirillov as a common generalization of Schubert polynomials, dual α-Grothendieck polynomials, Di Francesco–Zinn–Justin polynomials, etc. Then Kirillov conjectured that the coefficients of every generalized Hecke-Grothendieck polynomial are nonnegative combinations of certain parameters. Here we prove a weak version of Kirillov's conjecture, that is

• Balanced weighing matrices
J. Comb. Theory A (IF 1.192) Pub Date : 2021-10-20
Hadi Kharaghani, Thomas Pender, Sho Suda

A unified approach to the construction of weighing matrices and certain symmetric designs is presented. Assuming the weight p in a weighing matrix W(n,p) is an odd prime power, it is shown that there is aW(pm+1−1p−1(n−1)+1,pm+1) for each positive integer m. The case of n=p+1 reduces to the balanced weighing matrices with classical parametersW(pm+2−1p−1,pm+1). The equivalence with certain classes of

• A lower bound on the number of inequivalent APN functions
J. Comb. Theory A (IF 1.192) Pub Date : 2021-10-14
Christian Kaspers, Yue Zhou

In this paper, we establish a lower bound on the total number of inequivalent APN functions on the finite field with 22m elements, where m is even. We obtain this result by proving that the APN functions introduced by Pott and the second author , which depend on three parameters k, s and α, are pairwise inequivalent for distinct choices of the parameters k and s. Moreover, we determine the automorphism

• Skew-morphisms of nonabelian characteristically simple groups
J. Comb. Theory A (IF 1.192) Pub Date : 2021-10-01
Jiyong Chen, Shaofei Du, Cai Heng Li

A skew-morphism of a finite group G is a permutation σ on G fixing the identity element such that the product of 〈σ〉 with the left regular representation of G forms a permutation group on G. This permutation group is called the skew-product group of σ. The skew-morphism was introduced as an algebraic tool to investigate regular Cayley maps. In this paper, we characterize skew-products of skew-morphisms

• Gröbner fans of Hibi ideals, generalized Hibi ideals and flag varieties
J. Comb. Theory A (IF 1.192) Pub Date : 2021-09-29
Igor Makhlin

The main goal of this paper is to give explicit descriptions of two maximal cones in the Gröbner fan of the Plücker ideal. These cones correspond to the monomial ideals given by semistandard and PBW-semistandard Young tableaux. For the first cone, as an intermediate result we obtain the description of a maximal cone in the Gröbner fan of any Hibi ideal. For the second, we generalize the notion of Hibi

• Doubly transitive lines I: Higman pairs and roux
J. Comb. Theory A (IF 1.192) Pub Date : 2021-09-28
Joseph W. Iverson, Dustin G. Mixon

We study lines through the origin of finite-dimensional complex vector spaces that enjoy a doubly transitive automorphism group. In doing so, we make fundamental connections with both discrete geometry and algebraic combinatorics. In particular, we show that doubly transitive lines are necessarily optimal packings in complex projective space, and we introduce a fruitful generalization of regular abelian

• The Lemmens-Seidel conjecture and forbidden subgraphs
J. Comb. Theory A (IF 1.192) Pub Date : 2021-09-15
Meng-Yue Cao, Jack H. Koolen, Yen-Chi Roger Lin, Wei-Hsuan Yu

In this paper we show that the conjecture of Lemmens and Seidel of 1973 for systems of equiangular lines with common angle arccos⁡(1/5) is true. Our main tool is forbidden subgraphs for smallest Seidel eigenvalue −5.

• Koszul algebras and flow lattices
J. Comb. Theory A (IF 1.192) Pub Date : 2021-09-08
Zsuzsanna Dancso, Anthony M. Licata

We provide a homological algebraic realization of the lattices of integer cuts and integer flows of graphs. To a finite 2-edge-connected graph Γ with a spanning tree T, we associate a finite dimensional Koszul algebra AΓ,T. Under the construction, planar dual graphs with dual spanning trees are associated Koszul dual algebras. The Grothendieck group of the category of finitely-generated AΓ,T modules

• Sizes of simultaneous core partitions
J. Comb. Theory A (IF 1.192) Pub Date : 2021-09-03
Chaim Even-Zohar

There is a well-studied correspondence by Jaclyn Anderson between partitions that avoid hooks of length s or t and certain binary strings of length s+t. Using this map, we prove that the total size of a random partition of this kind converges in law to Watson's U2 distribution, as conjectured by Doron Zeilberger.

• A proof of the theta operator conjecture
J. Comb. Theory A (IF 1.192) Pub Date : 2021-09-02
Marino Romero

In the context of the (generalized) Delta Conjecture and its compositional form, D'Adderio, Iraci, and Vanden Wyngaerd recently stated a conjecture relating two symmetric function operators, Dk and Θk. We prove this Theta Operator Conjecture, finding it as a consequence of the five-term relation of Mellit and Garsia. As a result, we find surprising ways of writing the Dk operators. Even though we deal

• The stabilizing index and cyclic index of the coalescence and Cartesian product of uniform hypergraphs
J. Comb. Theory A (IF 1.192) Pub Date : 2021-09-01
Yi-Zheng Fan, Meng-Yu Tian, Min Li

Let G be connected uniform hypergraph and let A(G) be the adjacency tensor of G. The stabilizing index of G is exactly the number of eigenvectors of A(G) associated with the spectral radius, and the cyclic index of G is exactly the number of eigenvalues of A(G) with modulus equal to the spectral radius. Let G1⊙G2 and G1□G2 be the coalescence and Cartesian product of connected m-uniform hypergraphs

• Abelian closures of infinite binary words
J. Comb. Theory A (IF 1.192) Pub Date : 2021-08-27
Svetlana Puzynina, Markus A. Whiteland

Two finite words u and v are called Abelian equivalent if each letter occurs equally many times in both u and v. The abelian closure A(x) of (the shift orbit closure of) an infinite word x is the set of infinite words y such that, for each factor u of y, there exists a factor v of x which is abelian equivalent to u. The notion of an abelian closure gives a characterization of Sturmian words: among

• Compression of M♮-convex functions — Flag matroids and valuated permutohedra
J. Comb. Theory A (IF 1.192) Pub Date : 2021-08-25
Satoru Fujishige, Hiroshi Hirai

Murota (1998) and Murota and Shioura (1999) introduced concepts of M-convex function and M♮-convex function as discrete convex functions, which are generalizations of valuated matroids due to Dress and Wenzel (1992). In the present paper we consider a new operation defined by a convolution of sections of an M♮-convex function that transforms the given M♮-convex function to an M-convex function, which

• Combinatorial proof of Selberg's integral formula
J. Comb. Theory A (IF 1.192) Pub Date : 2021-08-23
Alexander M. Haupt

In this paper we present a combinatorial proof of Selberg's integral formula. We prove a theorem about the number of topological orderings of a certain related directed graph bijectively. Selberg's integral formula then follows by induction. This solves a problem posed by R. Stanley in 2009. Our proof is based on Anderson's analytic proof of the formula. As part of the proof we show a further generalisation

• Dyson's crank and the mex of integer partitions
J. Comb. Theory A (IF 1.192) Pub Date : 2021-08-19
Brian Hopkins, James A. Sellers, Dennis Stanton

Andrews and Newman have recently introduced the notion of the mex of a partition, the smallest positive integer that is not a part. The concept has been used since at least 2006, though, with connections to Frobenius symbols. Recently, the parity of the mex has been associated to the crank statistic named by Dyson in 1944. In this note, we extend and strengthen the connection between the crank and

• Universal singular exponents in catalytic variable equations
J. Comb. Theory A (IF 1.192) Pub Date : 2021-08-19
Michael Drmota, Marc Noy, Guan-Ru Yu

Catalytic equations appear in several combinatorial applications, most notably in the enumeration of lattice paths and in the enumeration of planar maps. The main purpose of this paper is to show that the asymptotic estimate for the coefficients of the solutions of (so-called) positive catalytic equations has a universal asymptotic behavior. In particular, this provides a rationale why the number of

• Lifting the dual immaculate functions
J. Comb. Theory A (IF 1.192) Pub Date : 2021-08-11
Sarah Mason, Dominic Searles

We introduce a reverse variant of the dual immaculate quasisymmetric functions, mirroring the dichotomy between quasisymmetric Schur functions and Young quasisymmetric Schur functions, and establish a lift of this basis to the polynomial ring. We show that taking stable limits of these reverse dual immaculate slide polynomials produces the reverse dual immaculate quasisymmetric functions, and we establish

• Refined Cauchy identity for spin Hall–Littlewood symmetric rational functions
J. Comb. Theory A (IF 1.192) Pub Date : 2021-08-06
Leonid Petrov

Fully inhomogeneous spin Hall–Littlewood symmetric rational functions Fλ arise in the context of sl(2) higher spin six vertex models, and are multiparameter deformations of the classical Hall–Littlewood symmetric polynomials. We obtain a refined Cauchy identity expressing a weighted sum of the product of two Fλ's as a determinant. The determinant is of Izergin–Korepin type: it is the partition function

• Iterated differences sets, Diophantine approximations and applications
J. Comb. Theory A (IF 1.192) Pub Date : 2021-08-06

Let v be an odd real polynomial (i.e. a polynomial of the form ∑j=1ℓajx2j−1). We utilize sets of iterated differences to establish new results about sets of the form R(v,ϵ)={n∈N|‖v(n)‖<ϵ} where ‖⋅‖ denotes the distance to the closest integer. We then apply the new Diophantine results to obtain applications to ergodic theory and combinatorics. In particular, we obtain a new characterization of weakly

• Cyclic permutations: Degrees and combinatorial types
J. Comb. Theory A (IF 1.192) Pub Date : 2021-08-06
Saeed Zakeri

This note will give an enumeration of n-cycles in the symmetric group Sn by their degree (also known as their cyclic descent number) and studies similar counting problems for the conjugacy classes of n-cycles under the action of the rotation subgroup of Sn. This is achieved by relating such cycles to periodic orbits of an associated dynamical system acting on the circle. We also compute the mean and

• On sum-of-tails identities
J. Comb. Theory A (IF 1.192) Pub Date : 2021-08-03
Rajat Gupta

In this article, a finite analogue of the generalized sum-of-tails identity of Andrews and Freitas is obtained. We derive several interesting results as special cases of this analogue, in particular, a recent identity of Dixit, Eyyunni, Maji and Sood. We derive a new extension of Abel's lemma with the help of which we obtain a one-parameter generalization of a sum-of-tails identity of Andrews, Garvan

• On strengthenings of the intersecting shadow theorem
J. Comb. Theory A (IF 1.192) Pub Date : 2021-08-02
P. Frankl, G.O.H. Katona

Let n>k>t≥j≥1 be integers. Let X be an n-element set, (Xk) the collection of its k-subsets. A family F⊂(Xk) is called t-intersecting if |F∩F′|≥t for all F,F′∈F. The j'th shadow ∂jF is the collection of all (k−j)-subsets that are contained in some member of F. Estimating |∂jF| as a function of |F| is a widely used tool in extremal set theory. A classical result of the second author (Theorem 1.3) provides

• Hypergraphs without exponents
J. Comb. Theory A (IF 1.192) Pub Date : 2021-07-29
Zoltán Füredi, Dániel Gerbner

A short, concise proof is given for that for k≥5 there exists a k-uniform hypergraph H without exponent, i.e., when the Turán function is not polynomial in n. More precisely, we have ex(n,H)=o(nk−1) but it exceeds nk−1−c for any positive c for n>n0(k,c). We conjecture that this is true for k∈{3,4} as well.

• On the combinatorics of string polytopes
J. Comb. Theory A (IF 1.192) Pub Date : 2021-07-29
Yunhyung Cho, Yoosik Kim, Eunjeong Lee, Kyeong-Dong Park

For a reduced word i of the longest element in the Weyl group of SLn+1(C), one can associate the string cone Ci which parametrizes the dual canonical bases. In this paper, we classify all i's such that Ci is simplicial. We also prove that for any regular dominant weight λ of sln+1(C), the corresponding string polytope Δi(λ) is unimodularly equivalent to the Gelfand–Cetlin polytope associated to λ if

• Primitive permutation IBIS groups
J. Comb. Theory A (IF 1.192) Pub Date : 2021-07-29
Andrea Lucchini, Marta Morigi, Mariapia Moscatiello

Let G be a finite permutation group on Ω. An ordered sequence of elements of Ω, (ω1,…,ωt), is an irredundant base for G if the pointwise stabilizer G(ω1,…,ωt) is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of G have the same size we say that G is an IBIS group. In this paper we show that if a primitive permutation group is IBIS, then it must be almost

• Novák's conjecture on cyclic Steiner triple systems and its generalization
J. Comb. Theory A (IF 1.192) Pub Date : 2021-07-29
Tao Feng, Daniel Horsley, Xiaomiao Wang

Novák conjectured in 1974 that for any cyclic Steiner triple systems of order v with v≡1(mod6), it is always possible to choose one block from each block orbit so that the chosen blocks are pairwise disjoint. We consider the generalization of this conjecture to cyclic (v,k,λ)-designs with 1⩽λ⩽k−1. Superimposing multiple copies of a cyclic symmetric design shows that the generalization cannot hold for

• Prime power variations of higher Lien modules
J. Comb. Theory A (IF 1.192) Pub Date : 2021-07-27
Sheila Sundaram

We define, for each subset S of the set P of primes, an Sn-module LienS with interesting properties. Lien∅ is the well-known representation Lien of Sn afforded by the free Lie algebra, while LienP is the module Conjn of the conjugacy action of Sn on n-cycles. For arbitrary S the module LienS interpolates between the representations Lien and Conjn. We consider the symmetric and exterior powers of LienS

• A semi-finite form of the quintuple product identity
J. Comb. Theory A (IF 1.192) Pub Date : 2021-07-26
Jun-Ming Zhu, Zhi-Zheng Zhang

The quintuple product identity is deduced from the q-Dixon formula.

• New constructions of strongly regular Cayley graphs on abelian non p-groups
J. Comb. Theory A (IF 1.192) Pub Date : 2021-07-26
Koji Momihara

Davis and Jedwab (1997) established a great construction theory unifying many previously known constructions of difference sets, relative difference sets and divisible difference sets. They introduced the concept of building blocks, which played an important role in the theory. On the other hand, Polhill (2010) gave a construction of Paley type partial difference sets (conference graphs) based on a

• Chromatic posets
J. Comb. Theory A (IF 1.192) Pub Date : 2021-07-15
Samantha Dahlberg, Adrian She, Stephanie van Willigenburg

In 1995 Stanley introduced the chromatic symmetric function XG of a graph G, whose e-positivity and Schur-positivity has been of large interest. In this paper we study the relative e-positivity and Schur-positivity between connected graphs on n vertices. We define and investigate two families of posets on distinct chromatic symmetric functions. The relations depend on the e-positivity or Schur-positivity

• Counting tanglegrams with species
J. Comb. Theory A (IF 1.192) Pub Date : 2021-07-09
Ira M. Gessel

A tanglegram is a pair of binary trees with the same set of leaves. Unlabeled tanglegrams were counted recently by Billey, Konvalinka, and Matsen, who also proposed the problem of counting several variations of unlabeled tanglegrams. We use the theory of combinatorial species to solve these problems.

• Induced and non-induced poset saturation problems
J. Comb. Theory A (IF 1.192) Pub Date : 2021-07-02
Balázs Keszegh, Nathan Lemons, Ryan R. Martin, Dömötör Pálvölgyi, Balázs Patkós

A subfamily G⊆F⊆2[n] of sets is a non-induced (weak) copy of a poset P in F if there exists a bijection i:P→G such that p≤Pq implies i(p)⊆i(q). In the case where in addition p≤Pq holds if and only if i(p)⊆i(q), then G is an induced (strong) copy of P in F. We consider the minimum number sat(n,P) [resp. sat⁎(n,P)] of sets that a family F⊆2[n] can have without containing a non-induced [induced] copy

• Young's seminormal basis vectors and their denominators
J. Comb. Theory A (IF 1.192) Pub Date : 2021-07-01
Ming Fang, Kay Jin Lim, Kai Meng Tan

We study Young's seminormal basis vectors of the dual Specht modules of the symmetric group, indexed by a certain class of standard tableaux, and their denominators. These vectors include those whose denominators control the splitting of the canonical morphism Δ(λ+μ)→Δ(λ)⊗Δ(μ) over Z(p), where Δ(ν) is the Weyl module of the classical Schur algebra labelled by ν.

• On the generic family of Cayley graphs of a finite group
J. Comb. Theory A (IF 1.192) Pub Date : 2021-06-30
Czesław Bagiński, Piotr Grzeszczuk

The Cayley graphs Gm(G)=Cay(Gm,S) of Cartesian powers of the finite group G with respect to canonical symmetric subsets S=S(m)⊂Gm are investigated. It is proved that for each m>1 the graph Gm(G) determines the group G up to isomorphism. The groups of automorphisms Aut(Gm(G)) are determined. It is shown that if G is a non-abelian group, then Aut(Gm(G))≃(Gm⋊Aut(G))⋊Dm+1, where Dm+1 is the dihedral group

• Chromatic symmetric function of graphs from Borcherds algebras
J. Comb. Theory A (IF 1.192) Pub Date : 2021-06-28
G. Arunkumar

The Weyl denominator identity has interesting combinatorial properties for several classes of Lie algebras. Along these lines, we prove that given a finite graph G, the chromatic symmetric function XG can be recovered from the Weyl denominator identity of a Borcherds-Kac-Moody Lie algebra g whose associated graph is G. This gives a connection between (a) the coefficients appearing when the chromatic

• On strong Sidon sets of integers
J. Comb. Theory A (IF 1.192) Pub Date : 2021-06-25
Yoshiharu Kohayakawa, Sang June Lee, Carlos Gustavo Moreira, Vojtěch Rödl

A set S⊂N of positive integers is a Sidon set if the pairwise sums of its elements are all distinct, or, equivalently, if|(x+w)−(y+z)|≥1 for every x,y,z,w∈S with x

• Increasing paths in countable graphs
J. Comb. Theory A (IF 1.192) Pub Date : 2021-06-23
Andrii Arman, Bradley Elliott, Vojtěch Rödl

In this paper we study variations of an old result by Müller, Reiterman, and the last author stating that a countable graph has a subgraph with infinite degrees if and only if in any labeling of the vertices (or edges) of this graph by positive integers one can always find an infinite increasing path. We study corresponding questions for simple hypergraphs and paths of infinite as well as of arbitrary

• A new partial geometry pg(5,5,2)
J. Comb. Theory A (IF 1.192) Pub Date : 2021-06-23

We construct a new partial geometry with parameters pg(5,5,2), not isomorphic to the partial geometry of van Lint and Schrijver.

• Rainbow matchings for 3-uniform hypergraphs
J. Comb. Theory A (IF 1.192) Pub Date : 2021-06-11
Hongliang Lu, Xingxing Yu, Xiaofan Yuan

Kühn, Osthus, and Treglown and, independently, Khan proved that if H is a 3-uniform hypergraph with n vertices, where n∈3Z and large, and δ1(H)>(n−12)−(2n/32), then H contains a perfect matching. In this paper, we show that for n∈3Z sufficiently large, if F1,…,Fn/3 are 3-uniform hypergraphs with a common vertex set and δ1(Fi)>(n−12)−(2n/32) for i∈[n/3], then {F1,…,Fn/3} admits a rainbow matching, i

• Independence polynomials and Alexander-Conway polynomials of plumbing links
J. Comb. Theory A (IF 1.192) Pub Date : 2021-06-03
A. Stoimenow

We use the Chudnovsky-Seymour Real Root Theorem for independence polynomials to obtain some statements about the coefficients and roots of the Alexander and Conway polynomial of some types of plumbing links, addressing conjectures of Fox, Hoste and Liechti.

• Statistics on multipermutations and partial γ-positivity
J. Comb. Theory A (IF 1.192) Pub Date : 2021-06-03
Zhicong Lin, Jun Ma, Philip B. Zhang

We prove that the enumerative polynomials of Stirling multipermutations by the statistics of plateaux, descents and ascents are partial γ-positive. Specialization of our result to the Jacobi-Stirling permutations confirms a recent partial γ-positivity conjecture due to Ma, Yeh and the second named author. Our partial γ-positivity expansion, as well as a combinatorial interpretation for the corresponding

• Plane partitions of shifted double staircase shape
J. Comb. Theory A (IF 1.192) Pub Date : 2021-05-28
Sam Hopkins, Tri Lai

We give a product formula for the number of shifted plane partitions of shifted double staircase shape with bounded entries. This is the first new example of a family of shapes with a plane partition product formula in many years. The proof is based on the theory of lozenge tilings; specifically, we apply the “free boundary” Kuo condensation due to Ciucu.

• On t-core and self-conjugate (2t − 1)-core partitions in arithmetic progressions
J. Comb. Theory A (IF 1.192) Pub Date : 2021-05-18
Kathrin Bringmann, Ben Kane, Joshua Males

We extend recent results of Ono and Raji, relating the number of self-conjugate 7-core partitions to Hurwitz class numbers. Furthermore, we give a combinatorial explanation for the curious equality 2sc7(8n+1)=c4(7n+2). We also conjecture that an equality of this shape holds if and only if t=4, proving the cases t∈{2,3,5} and giving partial results for t>5.

• Characterising the secant lines of Q(4,q), q even
J. Comb. Theory A (IF 1.192) Pub Date : 2021-05-11
Susan G. Barwick, Alice M.W. Hui, Wen-Ai Jackson, Jeroen Schillewaert

We show that a set A of lines in PG(4,q), q even, is the set of secant lines of a parabolic (non-singular) quadric if and only if A satisfies the following three conditions: (I) every point of PG(4,q) lies on 0,12q3 or q3 lines of A; (II) every plane of PG(4,q) contains 0, 12q(q+1) or q2 lines of A; and (III) every hyperplane of PG(4,q) contains 12q2(q2+1), 12q3(q+1) or 12q2(q+1)2 lines of A.

• Bumpless pipe dreams and alternating sign matrices
J. Comb. Theory A (IF 1.192) Pub Date : 2021-05-10
Anna Weigandt

In their work on the infinite flag variety, Lam, Lee, and Shimozono  introduced objects called bumpless pipe dreams and used them to give a formula for double Schubert polynomials. We extend this formula to the setting of K-theory, giving an expression for double Grothendieck polynomials as a sum over a larger class of bumpless pipe dreams. Our proof relies on techniques found in an unpublished

• Proof of a conjecture of Adamchuk
J. Comb. Theory A (IF 1.192) Pub Date : 2021-05-06
Guo-Shuai Mao

In this paper, we prove a congruence which contains a congruence conjectured by Adamchuk (OEIS A066796 in 2006, http://oeis.org/A066796). For any prime p≡1(mod3) and a∈Z+, we have∑k=123(pa−1)(2kk)≡0(modp2).

• Harmonic differential forms for pseudo-reflection groups I. Semi-invariants
J. Comb. Theory A (IF 1.192) Pub Date : 2021-05-06
Joshua P. Swanson, Nolan R. Wallach

We provide a type-independent construction of an explicit basis for the semi-invariant harmonic differential forms of an arbitrary pseudo-reflection group in characteristic zero. Equivalently, we completely describe the structure of the χ-isotypic components of the corresponding super coinvariant algebras in one commuting and one anti-commuting set of variables, for all linear characters χ. In type

• Projective embeddings of M‾0,n and parking functions
J. Comb. Theory A (IF 1.192) Pub Date : 2021-05-04
Renzo Cavalieri, Maria Gillespie, Leonid Monin

The moduli space M‾0,n may be embedded into the product of projective spaces P1×P2×⋯×Pn−3, using a combination of the Kapranov map |ψn|:M‾0,n→Pn−3 and the forgetful maps πi:M‾0,i→M‾0,i−1. We give an explicit combinatorial formula for the multidegree of this embedding in terms of certain parking functions of height n−3. We use this combinatorial interpretation to show that the total degree of the embedding

• The Norton algebra of a Q-polynomial distance-regular graph
J. Comb. Theory A (IF 1.192) Pub Date : 2021-05-04
Paul Terwilliger

We consider the Norton algebra associated with a Q-polynomial primitive idempotent of the adjacency matrix for a distance-regular graph. We obtain a formula for the Norton algebra product that we find attractive.

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