Search

We investigate the Castelnuovo-Mumford regularity and the multiplicity of the toric ring associated with a three-dimensional Ferrers diagram. In particular, in the rectangular case, we provide direct formulas for these two important invariants. Then, we compare these invariants for an accompanying pair of Ferrers diagrams under some mild conditions and bound the Castelnuovo-Mumford regularity for more general cases., Comment: Revised following the reviewer's suggestions. 27 pages and 6 figures. Comments are welcome

We develop a nonstandard approach to exploring polynomials associated with peaks and runs of permutations. With the aid of a context-free grammar, or a set of substitution rules, one can perform a symbolic calculus, and the computation often becomes rather simple. From a grammar it follows at once a system of ordinary differential equations for the generating functions. Utilizing a certain constant property, it is even possible to deduce a single equation for each generating function. To bring the grammar to a combinatorial setting, we find a labeling scheme for up-down runs of a permutation, which can be regarded as a refined property, or the differentiability in a certain sense, in contrast to the usual counting argument for the recurrence relation. The labeling scheme also exhibits how the substitution rules arise in the construction of the combinatorial structures. Consequently, polynomials on peaks and runs can be dealt with in two ways, combinatorially or grammatically. The grammar also serves as a guideline to build a bijection between permutations and increasing trees that maps the number of up-down runs to the number of nonroot vertices of even degree. This correspondence can be adapted to left peaks and exterior peaks, and the key step of the construction is called the reflection principle., Revised version, 25 pages, 15 figures, to appear in J. Algebraic Combin

In our previous work, we equipped quiver Grassmannians for nilpotent representations of the equioriented cycle with an action of an algebraic torus. We show here that the equivariant cohomology ring is acted upon by a product of symmetric groups and we investigate this permutation action via GKM techniques. In the case of (type A) flag varieties, or Schubert varieties therein, we recover Tymoczko’s results on permutation representations.

An (association) scheme is said to be Frobenius if it is the scheme of a Frobenius group. A scheme which has the same tensor of intersection numbers as some Frobenius scheme is said to be pseudofrobenius. We establish a necessary and sufficient condition for an imprimitive pseudofrobenius scheme to be Frobenius. We also prove strong necessary conditions for existence of an imprimitive pseudofrobenius scheme which is not Frobenius. As a byproduct, we obtain a sufficient condition for an imprimitive Frobenius group $G$ with abelian kernel to be determined up to isomorphism only by the character table of $G$. Finally, we prove that the Weisfeiler-Leman dimension of a circulant graph with $n$ vertices and Frobenius automorphism group is equal to $2$ unless $n\in \{p,p^2,p^3,pq,p^2q\}$, where $p$ and $q$ are distinct primes., 16 pages

We construct the first infinite families of locally 2-arc transitive graphs with the property that the automorphism group has two orbits on vertices and is quasiprimitive on exactly one orbit, of twisted wreath type. This work contributes to Giudici, Li and Praeger’s program for the classification of locally 2-arc transitive graphs by showing that the star normal quotient twisted wreath category also contains infinitely many graphs.

The Gelfand representation of $\mathcal{S}_n$ is the multiplicity-free direct sum of the irreducible representations of $\mathcal{S}_n$. In this paper, we use a result of Adin, Postnikov, and Roichman to find a recursive generating function for the Gelfand character. In order to find this generating function, we investigate descents of so-called $\lambda$-unimodal involutions.

In this note we report the classification of all symmetric 2-(36,15,6) designs that admit an automorphism of order 2 and their incidence matrices generate an extremal ternary self-dual code. It is shown that up to isomorphism, there exists only one such design, having a full automorphism group of order 24, and the ternary code spanned by its incidence matrix is equivalent to the Pless symmetry code., Comment: 12 pages

We prove that, if $\Gamma$ is a finite connected $3$-valent vertex-transitive, or $4$-valent vertex- and edge-transitive graph, then either $\Gamma$ is part of a well-understood family of graphs, or every non-identity automorphism of $\Gamma$ fixes at most $1/3$ of the edges. This answers a question proposed by Primo\v{z} Poto\v{c}nik and the third author., Comment: 14 pages

Rigid tree honeycombs were introduced by Knutson, Tao, and Woodward and they were shown by Dykema, Collins, Timotin, and the authors to be sums of extreme rigid honeycombs, with uniquely determined summands up to permutations. Two extreme rigid honeycombs are essentially the same if they have proportional exit multiplicities and, up to this identification, there are countably many equivalence classes of such honeycombs. We describe two ways to approach the enumeration of these equivalence classes. The first method produces a (finite) list of all rigid tree honeycombs of fixed weight by looking at the locking patterns that can be obtained from a certain quadratic Diophantine equation. The second method constructs arbitrary rigid tree honeycombs from rigid overlays of two rigid tree honeycombs with strictly smaller weights. This allows, in principle, for an inductive construction of all rigid tree honeycombs starting with those of unit weight. We also show that some rigid overlays of two rigid tree honeycombs give rise to an infinite sequence of rigid tree honeycombs of increasing complexity but with a fixed number of nonzero exit multiplicities. This last result involves a new inflation/deflation construction that also produces other infinite sequences of rigid tree honeycombs., Comment: 45 pages, many figures

We prove that for every integer $k\geq 1$, there exists a connected graph $H_k$ such that $v(H_k)=reg(H_k)+k$, where $v(G)$ and $reg(G)$ denote the $v$-number and the (Castelnuovo-Mumford) regularity of a graph $G$ respectively.

We study defining inequalities of string cones via a potential function on a reduced double Bruhat cell. We give a necessary criterion for the potential function to provide a minimal set of inequalities via tropicalization and conjecture an equivalence.

Let $\mu_{\infty}\subseteq\mathbb{C}$ be the collection of roots of unity and $\mathcal{C}_{n}:=\{(s_{1},\cdots,s_{n})\in\mu_{\infty}^{n}:s_{i}\neq s_{j}\text{ for any }1\leq i

We present the characteristic polynomial for the transition matrix of a vertex-face walk on a graph, and obtain its spectra. Furthermore, we express the characteristic polynomial for the transition matrix of a vertex-face walk on the 2-dimensional torus by using its adjacency matrix, and obtain its spectra. As an application, we define a new walk-type zeta function with respect to the transition matrix of a vertex-face walk on the 2-dimensional torus, and present its explicit formula., Comment: 14 pages. arXiv admin note: text overlap with arXiv:2103.12971, arXiv:2011.14162

Skew stable Grothendieck polynomials are $K$-theoretic analogues of skew Schur polynomials. We give a free-fermionic presentation of skew stable Grothendieck polynomials and their dual symmetric functions. By using our presentation, we derive a family of determinantal formulas, which are $K$-analogues of the Jacobi-Trudi formula for skew Schur functions. We also introduce a combinatorial method to calculate certain expansions of skew (dual) stable Grothendieck polynomials by using the non-commutative supersymmetric Schur functions., 25 pages. This work is a continuation of the author's previous work arXiv:1905.07692

This paper presents new results on the identities satisfied by the sylvester and Baxter monoids. We show how to embed these monoids, of any rank strictly greater than 2, into a direct product of copies of the corresponding monoid of rank 2. This confirms that all monoids of the same family, of rank greater than or equal to 2, satisfy exactly the same identities. We then give a complete characterization of those identities, and prove that the varieties generated by the sylvester and the Baxter monoids have finite axiomatic rank, by giving a finite basis for them., 29 pages. arXiv admin note: text overlap with arXiv:2010.06953

In the present paper, we study conic-line arrangements having nodes, tacnodes, and ordinary triple points as singularities. We provide combinatorial constraints on such arrangements and we give the complete classification of free arrangements in this class., 24 pages, 2 figures, a minor revision incorporating referee's remarks

We describe a new class of maximal cliques, with a vector space structure, of Cayley graphs defined on the additive group of a field. In particular, we show that in the cubic Paley graph with order $q^3$, the subfield with $q$ elements forms a maximal clique. Similar statements also hold for quadruple Paley graphs and Peisert graphs with quartic order., 12 pages, revised based on referees' suggestions

Tropical ideals are a class of ideals in the tropical polynomial semiring that combinatorially abstracts the possible collections of supports of all polynomials in an ideal over a field. We study zero-dimensional tropical ideals I with Boolean coefficients in which all underlying matroids are paving matroids, or equivalently, in which all polynomials of minimal support have support of size deg(I) or deg(I)+1 -- we call them paving tropical ideals. We show that paving tropical ideals of degree d+1 are in bijection with $\mathbb Z^n$-invariant d-partitions of $\mathbb Z^n$. This implies that zero-dimensional tropical ideals of degree 3 with Boolean coefficients are in bijection with $\mathbb Z^n$-invariant 2-partitions of quotient groups of the form $\mathbb Z^n/L$. We provide several applications of these techniques, including a construction of uncountably many zero-dimensional degree-3 tropical ideals in one variable with Boolean coefficients, and new examples of non-realizable zero-dimensional tropical ideals., Comment: 13 pages

Let $G$ be a graph with vertex set $V=\{v_{1},\dots,v_{n}\}$ and adjacency matrix $A.$ For a subset $S$ of $V$ let $\e=(x_{1},\,\dots,\,x_{n})^{\tt T}$ be the characteristic vector of $S,$ that is, $x_{\ell}=1$ if $v_{\ell}\in S$ and $x_{\ell}=0$ otherwise. Then the $n\times n$ matrix $$W^{S}:=\big[{\rm e},\,A{\rm e},\,A^{2}{\rm e},\dots,A^{n-1}{\rm e}\big]$$ is the {\it walk matrix} of $G$ for $S.$ This name relates to the fact that in $W^{S}$ the $k^{\rm th}$ entry in the row corresponding to $v_{\ell}$ is the number of walks of length $k-1$ from $v_{\ell}$ to some vertex in $S$. Since $A$ is symmetric the characteristic vector of $S$ can be written uniquely as a sum of eigenvectors of $A.$ In particular, we may enumerate the distinct eigenvalues $��_{1},\dots, ��_{s}$ of $A$ so that \begin{eqnarray}\label{SSA}{\rm SD}(S)\!:\,\e&=&\e_{1}+\e_{2}+\dots+\e_{r}\, \end{eqnarray} where $r\leq s$ and $\e_{i}$ is an eigenvector of $A$ of $��_{i}$ for all $1\leq i\leq r. We refer to (\ref{SSA}) as the {\it spectral decomposition} of $S,$ or more properly, of its characteristic vector. The key result of this paper is that the walk matrix $W^{S}$ determines the spectral decomposition of $S$ and {\it vice versa.} This holds for any non-empty set $S$ of vertices of the graph and explicit algorithms which establish this correspondence are given. In particular, we show that the number of distinct eigenvectors that appear in \,(\ref{SSA})\, is equal to the rank of $W^{S}.$ Several theorems can be derived from this result. We show that $W^{S}$ determines the adjacency matrix of $G$ if $W^{S}$ has rank $\geq n-1$. This theorem is best possible as there are examples of pairs of graphs with the same walk matrix of rank $n-2$ but with different adjacency matrices., 28 pages, 7 figures

In this article, we investigate $2$-$(v,k,\lambda)$ designs with $\gcd(r,\lambda)=1$ admitting flag-transitive automorphism groups $G$. We prove that if $G$ is an almost simple group, then such a design belongs to one of the seven infinite families of $2$-designs or it is one of the eleven well-known examples. We describe all these examples of designs. We, in particular, prove that if $\mathcal{D}$ is a symmetric $(v,k,\lambda)$ design with $\gcd(k,\lambda)=1$ admitting a flag-transitive automorphism group $G$, then either $G\leq A\Gamma L_{1}(q)$ for some odd prime power $q$, or $\mathcal{D}$ is a projective space or the unique Hadamard design with parameters $(11,5,2)$., Comment: arXiv admin note: text overlap with arXiv:1904.10518

Let $\mathcal{D}$ be a weighted oriented graph and let $I(\mathcal{D})$ be its edge ideal in a polynomial ring $R$. We give the formula of Castelnuovo-Mumford regularity of $R/I(\mathcal{D})$ when $\mathcal{D}$ is a weighted oriented path or cycle such that edges of $\mathcal{D}$ are oriented in one direction. Additionally, we compute the projective dimension for this class of graphs., 22 pages, 5 figures

Given a lattice polygon $P$ with $g$ interior lattice points, we associate to it the moduli space of tropical curves of genus $g$ with Newton polygon $P$. We completely classify the possible dimensions such a moduli space can have. For non-hyperelliptic polygons the dimension must be between $g$ and $2g+1$, and can take on any integer value in this range, with exceptions only in the cases of genus $3$, $4$, and $7$. We provide a similar result for hyperelliptic polygons, for which the range of dimensions is from $g$ to $2g-1$. In the case of non-hyperelliptic polygons, our results also hold for the moduli space of algebraic curves that are non-degenerate with respect to $P$., Comment: 17 pages, 15 figures

This is a post-peer-review, pre-copyedit version of an article published in Journal of algebraic combinatorics. The final authenticated version is available online at: http://dx.doi.org/10.1007/s10801-021-01049-3 We describe, in a very explicit way, a method for determining the spectra and bases of all the corresponding eigenspaces of arbitrary lifts of graphs (regular or not). The first author has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 734922. The research of C. Dalf´o and M. A. Fiol has been partially supported by AGAUR from the Catalan Government under project 2017SGR1087 and by MICINN from the Spanish Government under project PGC2018-095471-B-I00. The research of C. Dalf´o has also been supported by MICINN from the Spanish Government under project MTM2017-83271-R. The authors S. Pavl´ıkov´a and J. Sir´aˇn acknowledge support from the ˇ APVV Research Grants 15-0220 and 17-0428, and the VEGA Research Grants 1/0142/17 and 1/0238/19.

We prove a result on the asymptotic proportion of randomly chosen pairs of permutations in the symmetric group $S_n$ which "invariably" generate a nonsolvable subgroup, i.e., whose cycle structures cannot possibly both occur in the same solvable subgroup of $S_n$. As an application, we obtain that for a large degree "random" integer polynomial $f$, reduction modulo two different primes can be expected to suffice to prove the nonsolvability of $Gal(f/\mathbb{Q})$., Fixed some error in the introduction, compared to Version 1

Let $$\mathcal {P}$$ be a simple thin polyomino, namely a polyomino that has no holes and does not contain a square tetromino as a subpolyomino. In this paper, we determine the reduced Hilbert–Poincare series $$h(t)/(1-t)^d$$ of $$K[\mathcal {P}]$$ by proving that h(t) is the rook polynomial of $$\mathcal {P}$$ . As an application, we characterize the Gorenstein simple thin polyominoes.

An iterative formula for the Kostka-Foulkes polynomials is given using the vertex operator realization of the Hall-Littlewood polynomials. The operational formula can handle large Kostka-Foulkes polynomials, and a stability property for the Kostka-Foulkes polynomials is shown. We also use our algorithm to give a formula of $K_{\lambda\mu}(t)$ for $\mu$ being hook-shaped., Comment: 10 pages; final version (revised version with new title. New hook-shaped formula and stability property are added)

In this paper, a question due to Heckenberger, Shareshian and Welker on racks in Heckenberger et al. (Trans Am Math Soc, 372:1407–1427, 2019) is positively answered. A rack is a set together with a self-distributive bijective binary operation. We show that the lattice of subracks of every finite rack is complemented. Moreover, we characterize finite modular lattices of subracks in terms of complements of subracks. Also, we introduce a certain class of racks including all finite groups with the conjugation operation, called G-racks, and we study some of their properties. In particular, we show that a finite G-rack has the homotopy type of a sphere. Further, we show that the lattice of subracks of an infinite rack is not necessarily complemented which gives an affirmative answer to the aforementioned question. Indeed, we show that the lattice of subracks of the set of rational numbers, as a dihedral rack, is not complemented. Finally, we show that being a Boolean algebra, pseudocomplemented and uniquely complemented as well as distributivity are equivalent for the lattice of subracks of a rack.

In a 1989 paper \cite{arasu2}, Arasu used an observation about multipliers to show that no $(352,27,2)$ difference set exists in any abelian group. The proof is quite short and required no computer assistance. We show that it may be applied to a wide range of parameters $(v,k,\lambda)$, particularly for small values of $\lambda$. With it a computer search was able to show that the Prime Power Conjecture is true up to order $2 \cdot 10^{10}$, extend Hughes and Dickey's computations for $\lambda=2$ and $k \leq 5000$ up to $10^{10}$, and show nonexistence for many other parameters., Comment: 8 pages

Gathering different results from singularity theory, geometry and combinatorics, we show that the spectrum at infinity of a tame Laurent polynomial counts lattice points in polytopes and we deduce an effective algorithm in order to compute the Ehrhart polynomial of a simplex containing the origin as an interior point., References updated. Two remarks on the distribution (unimodality, variance) of the spectrum at infinity added

We consider the problem of finding a Young diagram minimizing the sum of evaluations of a given pair of functions on the parts of the associated pair of conjugate partitions. While there are exponentially many diagrams, we show it is polynomial time solvable.

The notion of disjoint weighing matrices is introduced as a generalization of orthogonal designs. A recursive construction along with a computer search lead to some infinite classes of disjoint weighing matrices, which in turn are shown to form commutative association schemes with 3 or 4 classes., Comment: 11 pages

A signed graph is a pair $$(G,\tau )$$ of a graph G and its sign $$\tau $$ , where a sign $$\tau $$ is a function from $$\{ (e,v)\mid e\in E(G),v\in V(G), v\in e\}$$ to $$\{1,-1\}$$ . Note that graphs or digraphs are special cases of signed graphs. In this paper, we study the toric ideal $$I_{(G,\tau )}$$ associated with a signed graph $$(G,\tau )$$ , and the results of the paper give a unified idea to explain some known results on the toric ideals of a graph or a digraph. We characterize all primitive binomials of $$I_{(G,\tau )}$$ and then focus on the complete intersection property. More precisely, we find a complete list of graphs G such that $$I_{(G,\tau )}$$ is a complete intersection for every sign $$\tau $$ .

Let $G$ be a simple graph on $n$ vertices and $J_G$ denote the corresponding binomial edge ideal in $S = K[x_1, \ldots, x_n, y_1,\ldots, y_n].$ We prove that the Castelnuovo-Mumford regularity of $J_G$ is bounded above by $c(G)+1$ when $G$ is a quasi-block graph or semi-block graph. We give another proof of Saeedi Madani-Kiani regularity upper bound conjecture for chordal graphs. We obtain the regularity of binomial edge ideals of Jahangir graphs. Later, we establish a sufficient condition for Hibi-Matsuda conjecture to be true., Comment: 14 pages

A shelling of a graph, viewed as an abstract simplicial complex that is pure of dimension 1, is an ordering of its edges such that every edge is adjacent to some other edges appeared previously. In this paper, we focus on complete bipartite graphs and trees. For complete bipartite graphs, we obtain an exact formula for their shelling numbers. And for trees, we relate their shelling numbers to linear extensions of tree posets and bound shelling numbers using vertex degrees and diameter., Comment: 18 pages

We consider some combinatorial problems on matrix polynomials over finite fields. Using results from control theory we give a proof of a result of Helmke, Jordan and Lieb on the number of linear unimodular matrix polynomials over a finite field. As an application of our results we give a new proof of a theorem of Chen and Tseng which answers a question of Niederreiter on splitting subspaces. We use our results to affirmatively resolve a conjecture on the probability that a matrix polynomial is unimodular., 14 pages

We show that the category of graphs has the structure of a 2-category with homotopy as the 2-cells. We then develop an explicit description of homotopies for finite graphs, in terms of what we call `spider moves'. We then create a category by modding out by the 2-cells of our 2-category, and use the spider moves to show that for finite graphs, this category is a homotopy category in the sense that it satisfies the universal property for localizing homotopy equivalences. We then show that finite stiff graphs form a skeleton of this homotopy category.

We introduce the notion of tropical curves of hyperelliptic type. These are tropical curves whose Jacobian is isomorphic to that of a hyperelliptic tropical curve, as polarized tropical abelian varieties. We show that this property depends only on the underlying graph of a tropical curve and is preserved when passing to genus $\geq 2$ connected minors. The main result is an excluded minors characterization of tropical curves of hyperelliptic type., Comment: 12 pages, 6 figures

We show that every finitely generated group G with an element of order at least $(5rank(G))^{12}$ admits a locally finite directed Cayley graph with automorphism group equal to G. If moreover G is not generalized dihedral, then the above Cayley directed graph does not have bigons. On the other hand, if G is neither generalized dicyclic nor abelian and has an element of order at least $(2rank(G))^{36}$, then it admits an undirected Cayley graph with automorphism group equal to G. This extends classical results for finite groups and free products of groups. The above results are obtained as corollaries of a stronger form of rigidity which says that the rigidity of the graph can be observed in a ball of radius 1 around a vertex. This strong rigidity result also implies that the Cayley (di)graph covers very few (di)graphs. In particular, we obtain Cayley graphs of Tarski monsters which essentially do not cover other quasi-transitive graphs. We also show that a finitely generated group admits a locally finite labelled unoriented Cayley graph with automorphism group equal to itself if and only if it is neither generalized dicyclic nor abelian with an element of order greater than 2., Comment: v2: 30 pages. Introduction and first section completely rewritten to improve the exposition. The bounds in Theorem 1 and Theorem 6 have been improved

We conjecture an explicit positive combinatorial formula for the expansion of unicellular LLT polynomials in the elementary symmetric basis. This is an analogue of the Shareshian-Wachs conjecture and previously studied by Panova and the author in 2018. We show that the conjecture for unicellular LLT polynomials implies a similar formula for vertical-strip LLT polynomials. We prove positivity in the elementary basis in for the class of graphs called `melting lollipops' previously considered by Huh, Nam and Yoo. This is done by proving a curious relationship between a generalization of charge and orientations of unit-interval graphs. We also provide short bijective proofs of Lee's three-term recurrences for unicellular LLT polynomials and we show that these recurrences are enough to generate all unicellular LLT polynomials associated with abelian area sequences., Minor edits, updated references

In the present paper, we investigate the complexity of infinite family of graphs $H_n=H_n(G_1,\,G_2,\ldots,G_m)$ obtained as a circulant foliation over a graph $H$ on $m$ vertices with fibers $G_{1},\,G_{2},\ldots,G_{m}.$ Each fiber $G_{i}=C_{n}(s_{i,1},\,s_{i,2},\ldots,s_{i,k_{i}})$ of this foliation is the circulant graph on $n$ vertices with jumps $s_{i,1},\,s_{i,2},\ldots,s_{i,k_{i}}.$ This family includes the family of generalized Petersen graphs, $I$-graphs, sandwiches of circulant graphs, discrete torus graphs and others. We obtain a closed formula for the number $\tau(n)$ of spanning trees in $H_{n}$ in terms of Chebyshev polynomials, investigate some arithmetical properties of this function and find its asymptotics as $n\to\infty.$, Comment: 14 pages

We derive some new signed Mahonian polynomials over the complex reflection group $G(r,1,n)=C_r\wr\mathfrak{S}_n$, where the "sign" is taken to be any of the $2r$ $1$-dim characters and the "Mahonian" statistics are the $\mathsf{lmaj}$ defined by Bagno and the $\mathsf{sor}$ defined by Eu et al. Various new signed Mahonian polynomials over Coxeter groups of types $B_n$ and $D_n$ are derived as well. We also investigate the signed counting polynomials on $G(r,1,n)$ for those statistics with the distribution $[r]_q[2r]_q\cdots [nr]_q$., 22 pages, 1 figure

Let $\mathfrak g$ be a simple Lie algebra with a Borel subalgebra $\mathfrak b$. Let $\Delta^+$ be the corresponding (po)set of positive roots and $\theta$ the highest root. A pair $\{\eta,\eta'\}\subset \Delta^+$ is said to be glorious, if $\eta,\eta'$ are incomparable and $\eta+\eta'=\theta$. Using the theory of abelian ideals of $\mathfrak b$, we (1) establish a relationship of $\eta,\eta'$ to certain abelian ideals associated with long simple roots, (2) provide a natural bijection between the glorious pairs and the pairs of adjacent long simple roots (i.e., some edges of the Dynkin diagram), and (3) point out a simple transform connecting two glorious pairs corresponding to the incident edges in the Dynkin digram. In types ${\bf DE}$, we prove that if $\{\eta,\eta'\}$ corresponds to the edge through the branching node of the Dynkin diagram, then the meet $\eta\wedge\eta'$ is the unique maximal non-commutative root. There is also an analogue of this property for all other types except type ${\bf A}$. As an application, we describe the minimal non-abelian ideals of $\mathfrak b$., Comment: 20 pages

We study the problem of counting real simple rational functions $\varphi$ with prescribed ramification data (i.e. a particular class of oriented real Hurwitz numbers of genus $0$). We introduce a signed count of such functions that is invariant under change of the branch locus, thus providing a lower bound for the actual count (which does depend on such change). We prove (non-)vanishing theorems for these signed counts and study their asymptotic growth when adding further simple branch points. The approach is based on the works of Itenberg and Zvonkine (arXiv:1609.05219) which treat the polynomial case., Comment: 34 pages, 14 figures, 4 tables

For an arbitrary finite permutation group $G$, subgroup of the symmetric group $S_\ell$, we determine the permutations involving only members of $G$ as $\ell$-patterns, i.e., avoiding all patterns in the set $S_\ell \setminus G$. The set of all $n$-permutations with this property constitutes again a permutation group. We consequently refine and strengthen the classification of sets of permutations closed under pattern involvement and composition that is due to Atkinson and Beals., Comment: 39 pages; extensively reorganized and rewritten, misprints corrected

In this paper, we study the class of free hyperplane arrangements. Specifically, we investigate the relations between freeness over a field of finite characteristic and freeness over $$\mathbb {Q}$$ .

Cumulants are a notion that comes from the classical probability theory, they are an alternative to a notion of moments. We adapt the probabilistic concept of cumulants to the setup of a linear space equipped with two multiplication structures. We present an algebraic formula which involves those two multiplications as a sum of products of cumulants. In our approach, beside cumulants, we make use of standard combinatorial tools as forests and their colourings. We also show that the resulting statement can be understood as an analogue of Leonov--Shiraev's formula. This purely combinatorial presentation leads to some conclusions about structure constant of Jack characters., 26 pages

Using the Poisson formula for resultants, and variants of the chip-firing game on graphs, we provide a combinatorial method for computing a class of resultants corresponding to the characteristic polynomials of the adjacency tensors of starlike hypergraphs including hyperpaths and hyperstars, which are given recursively and explicitly.

Erd\H{o}s and Pach (1983) introduced the natural degree-based generalisations of Ramsey numbers, where instead of seeking large monochromatic cliques in a $2$-edge coloured complete graph, we seek monochromatic subgraphs of high minimum or average degree. Here we expand the study of these so-called quasi-Ramsey numbers in a few ways, in particular, to multiple colours and to uniform hypergraphs. Quasi-Ramsey numbers are known to exhibit a certain unique phase transition and we show that this is also the case across the settings we consider. Our results depend on a density-biased notion of hypergraph discrepancy optimised over sets of bounded size, which may be of independent interest., Comment: 14 pages; some minor changes suggest by a referee. Accepted in Journal of Combinatorics

In this note we consider two configurations of twelve lines with nineteen triple points (i.e., points where three lines meet). Both of them have the same combinatorial features. In both configurations nine of twelve lines have five triple points and one double point, and the remaining three lines have four triple points and three double points. Taking the ideal of the triple points of these configurations we discover that, quite surprisingly, for one of the configurations the containment $I^{(3)} \subset I^2$ holds, while for the other it does not. Hence for ideals of points defined by configurations of lines the (non)containment of a symbolic power in an ordinary power is not determined alone by combinatorial features of the arrangement. Moreover, for the configuration with the non-containment $I^{(3)} \nsubseteq I^2$ we examine its parameter space, which turns out to be a rational curve, and thus establish the existence of a rational non-containment configuration of points. Such rational examples are very rare., Comment: 7 pages

In this paper, we define the combinatorial wall-crossing transformation and the generalized column regularization on partitions and prove that a certain composition of these two transformations has the same effect on the one-row partition $(n)$. As corollaries we explicitly describe the quotients of the partitions which arise in this process. We also prove that the one-row partition is the unique partition that stays regular at any step of the wall-crossing transformation.