There exists a variety of coloring problems for plane graphs, involving vertices, edges, and faces in all possible combinations. For instance, in the \emph{entire coloring} of a plane graph we are to color these three sets so that any pair of adjacent or incident elements get different colors. We study here some problems of this type from algebraic perspective, focusing on the \emph{facial} variant. We obtain several results concerning the \emph{Alon-Tarsi number} of various graphs derived from plane embeddings. This allows for extensions of some previous results for \emph{choosability} of these graphs to the game theoretic variant, know as \emph{paintability}. For instance, we prove that every plane graph is facially entirely \emph{$8$-paintable}, which means (metaphorically) that even a color-blind person can facially color the entire graph form lists of size $8$.
Nice formulae for plane partitions with bounded size of parts (or boxed plane partitions), which generalize the norm-trace generating function by Stanley and the trace generating function by Gansner, are exhibited. The derivation of the nice formulae is based on lattice path combinatorics of biorthogonal polynomials, especially of the little $q$-Laguerre polynomials and a generalization of the little $q$-Laguerre polynomials. A summation formula which generalizes the $q$-Chu-Vandermonde identity is also shown and utilized to prove the orthogonality of the generalized little $q$-Laguerre polyno…
The original motivation for study for hook length polynomials was to find a combinatorial proof for a hook length formula for binary trees given by Postnikov, as well as a proof for a hook length polynomial formula conjectured by Lascoux. In this paper, we define the hook length polynomial for plane forests of a given degree sequence type and show it can be factored into a product of linear forms. Some other enumerative results on forests are also given.
Motivated by a formula of A. Postnikov relating binary trees, we define the hook length polynomials for m-ary trees and plane forests, and show that these polynomials have a simple binomial expression. An integer value of this expression is C_{k,m}(n)=\frac{1}{mn+1}{(mn+1)k \choose n}, which we call the (k,m)-Catalan number. For proving the hook length formulas, we also introduce a combinatorial family, (k,m)-ary trees, which are counted by the (k,m)-Catalan numbers.
We establish a relation between the Bollobas-Riordan polynomial of a ribbon graph with the relative Tutte polynomial of a plane graph obtained from the ribbon graph using its projection to the plane in a nontrivial way. Also we give a duality formula for the relative Tutte polynomial of dual plane graphs and an expression of the Kauffman bracket of a virtual link as a specialization of the relative Tutte polynomial.
Generalizing a sequence of Lambert, Cayley and Ramanujan, Chapoton has recently introduced a polynomial sequence Q_n:=Q_n(x,y,z,t) defined by Q_1=1, Q_{n+1}=[x+nz+(y+t)(n+y\partial_y)]Q_n. In this paper we prove Chapoton's conjecture on the duality formula: Q_n(x,y,z,t)=Q_n(x+nz+nt,y,-t,-z), and answer his question about the combinatorial interpretation of Q_n. Actually we give combinatorial interpretations of these polynomials in terms of plane trees, half-mobile trees, and forests of plane trees. Our approach also leads to a general formula that unifies several known results for enumerating …
We define the disposition polynomial $R_{m}(x_1, x_2, ..., x_n)$ as $\prod_{k=0}^{m-1}(x_1+x_2+...+x_n+k)$. When $m=n-1$, this polynomial becomes the generating function of plane trees with respect to certain statistics as given by Guo and Zeng. When $x_i=1$ for $1\leq i\leq n$, $R_{m}(x_1, x_2, ..., x_n)$ reduces to the rising factorial $n(n+1)... (n+m-1)$. Guo and Zeng asked the question of finding a combinatorial proof of the formula for the generating function of plane trees with respect to the number of younger children and the number of elder children. We find a combinatorial interpretat…
Let $G$ be a simple graph of order n. The domination polynomial of G is the polynomial D(G,x) =\sum d(G, i)x^i, where d(G,i) is the number of dominating sets of G of size i. Every root of D(G,x) is called the domination root of G. It is clear that (0,\infty) is zero free interval for domination polynomial of a graph. It is interesting to investigate graphs which have complex domination roots with positive real parts. In this paper, we first investigate complexity of the domination polynomial at specific points. Then we present and investigate some families of graphs whose complex domination ro…
A polynomial with exactly two critical values is called a generalized Chebyshev polynomial. A polynomial with exactly three critical values is called a Zolotarev polynomial. Two Chebyshev polynomials $f$ and $g$ are called Z-homotopic, if there exists a family $p_α$, $α\in [0,1]$, where $p_0=f$, $p_1=g$ and $p_α$ is a Zolotarev polynomial, if $α\in (0,1)$. As each Chebyshev polynomial defines a plane tree (and vice versa), Z-homotopy can be defined for plane trees. In this work we prove some necessary geometric conditions for plane trees Z-homotopy, describe Z-homotopy for trees with 5 and 6 e…
We extend the Penrose polynomial, originally defined only for plane graphs, to graphs embedded in arbitrary surfaces. Considering this Penrose polynomial of embedded graphs leads to new identities and relations for the Penrose polynomial which can not be realized within the class of plane graphs. In particular, by exploiting connections with the transition polynomial and the ribbon group action, we find a deletion-contraction-type relation for the Penrose polynomial. We relate the Penrose polynomial of an orientable checkerboard colourable graph to the circuit partition polynomial of its media…
Inequalities are important features in the context of sequences of numbers and polynomials. The Bessenrodt--Ono inequality for partition numbers and Nekrasov--Okounkov polynomials has only recently been discovered. In this paper we study the log-concavity (Turán inequality) and Bessenrodt--Ono inequality for plane partitions and their polynomization.
A polynomial f is said to have the half-plane property if there is an open half-plane H, whose boundary contains the origin, such that f is non-zero whenever all the variables are in H. This paper answers several open questions regarding multivariate polynomials with the half-plane property and matroid theory. * We prove that the support of a multivariate polynomial with the half-plane property is a jump system. This answers an open question posed by Choe, Oxley, Sokal and Wagner and generalizes their recent result claiming that the same is true whenever the polynomial is also homogeneous. * W…
The discriminant of a polynomial map is central to problems from affine geometry and singularity theory. Standard methods for characterizing it rely on elimination techniques that can often be ineffective. This paper concerns polynomial maps on the two-dimensional torus defined over a field of Puiseux series. We present a combinatorial procedure for computing the tropical curve of the discriminant of maps determined by generic polynomials with given supports. Our results enable one to compute the Newton polytope of the discriminant of complex polynomial maps on the plane.
The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size, and its roots are called {\em independence roots}. We investigate the stability of such polynomials, that is, conditions under which the roots lie in the left half-plane (all of the real roots of independence polynomial are negative and hence lie in this half-plane). We show stability for all independence polynomials of graphs with independence number at most three, but for larger independence number we show that the independence polynomials can have roots arbitrarily far to the…
We generalize the recent work of S. Fomin and G. Mikhalkin on polynomial formulas for Severi degrees. The degree of the Severi variety of plane curves of degree d and delta nodes is given by a polynomial in d, provided delta is fixed and d is large enough. We extend this result to generalized Severi varieties parametrizing plane curves which, in addition, satisfy tangency conditions of given orders with respect to a given line. We show that the degrees of these varieties, appropriately rescaled, are given by a combinatorially defined "relative node polynomial" in the tangency orders, provided …
In this paper we investigate Erdős-Ko-Rado theorems in ovoidal circle geometries. We prove that in Möbius planes of even order greater than 2, and ovoidal Laguerre planes of odd order, the largest families of circles which pairwise intersect in at least one point, consist of all circles through a fixed point. In ovoidal Laguerre planes of even order, a similar result holds, but there is one other type of largest family of pairwise intersecting circles. As a corollary, we prove that the largest families of polynomials over $\mathbb F_q$ of degree at most $k$, with $2 \leq k < q$, which pairwise…
The cyclic sieving phenomenon of Reiner, Stanton, and White says that we can often count the fixed points of elements of a cyclic group acting on a combinatorial set by plugging roots of unity into a polynomial related to this set. One of the most impressive instances of the cyclic sieving phenomenon is a theorem of Rhoades asserting that the set of plane partitions in a rectangular box under the action of promotion exhibits cyclic sieving. In Rhoades's result the sieving polynomial is the size generating function for these plane partitions, which has a well-known product formula due to MacMah…
In this note, by counting some colored plane trees we obtain several binomial identities. These identities can be viewed as specific evaluations of certain generalizations of the Narayana polynomials. As consequences, it provides combinatorial proofs for a bijective problem in Stanley's collection "Bijective Proof Problems", a new formula for the Narayana polynomials as well as a new expression for the Harer-Zagier formula enumerating unicellular maps, in a unified way. Furthermore, we identify a class of plane trees, whose enumeration is closely connected to the Schröder numbers. Many other b…
The P-Eulerian polynomial counts the linear extensions of a labeled partially ordered set, P, by their number of descents. It is known that the P-Eulerian polynomials are real-rooted for various classes of posets P. The purpose of this paper is to extend these results to polynomials in several variables. To this end we study multivariate extensions of P-Eulerian polynomials and prove that for certain posets these polynomials are stable, i.e., non-vanishing whenever all variables are in the upper half-plane of the complex plane. A natural setting for our proofs is the Malvenuto-Reutenauer algeb…
According to the Göttsche conjecture (now a theorem), the degree N^{d, delta} of the Severi variety of plane curves of degree d with delta nodes is given by a polynomial in d, provided d is large enough. These "node polynomials" N_delta(d) were determined by Vainsencher and Kleiman-Piene for delta <= 6 and delta <= 8, respectively. Building on ideas of Fomin and Mikhalkin, we develop an explicit algorithm for computing all node polynomials, and use it to compute N_delta(d) for delta <= 14. Furthermore, we improve the threshold of polynomiality and verify Göttsche's conjecture on the optimal th…
The cutting plane approach to optimal matchings has been discussed by several authors over the past decades (e.g., Padberg and Rao '82, Grotschel and Holland '85, Lovasz and Plummer '86, Trick '87, Fischetti and Lodi '07) and its convergence has been an open question. We give a cutting plane algorithm that converges in polynomial-time using only Edmonds' blossom inequalities; it maintains half-integral intermediate LP solutions supported by a disjoint union of odd cycles and edges. Our main insight is a method to retain only a subset of the previously added cutting planes based on their dual v…
A polynomial P in n complex variables is said to have the "half-plane property" (or Hurwitz property) if it is nonvanishing whenever all the variables lie in the open right half-plane. Such polynomials arise in combinatorics, reliability theory, electrical circuit theory and statistical mechanics. A particularly important case is when the polynomial is homogeneous and multiaffine: then it is the (weighted) generating polynomial of an r-uniform set system. We prove that the support (set of nonzero coefficients) of a homogeneous multiaffine polynomial with the half-plane property is necessarily …
In this paper we prove that the generating series of the Poincare polynomials of quasihomogeneous Hilbert schemes of points in the plane has a beautiful decomposition into an infinite product. We also compute the generating series of the numbers of quasihomogeneous components in a moduli space of sheaves on the projective plane. The answer is given in terms of characters of the affine Lie algebra $\hat{sl}_m$.
Suppose $P$ is a symmetric convex polygon in the plane. We give a polynomial time algorithm that decides if $P$ can tile the plane by transations at some level (not necessarily at level one; this is multiple tiling). The main technical contribution is a polynomial time algorithm that selects, if this is possible, for each $j=1,2,\ldots,n$ one of two given vectors $e_j$ or $τ_j$ so that the selection spans a discrete additive subgroup.
We study the problem of approximating the value of the matching polynomial on graphs with edge parameter $γ$, where $γ$ takes arbitrary values in the complex plane. When $γ$ is a positive real, Jerrum and Sinclair showed that the problem admits an FPRAS on general graphs. For general complex values of $γ$, Patel and Regts, building on methods developed by Barvinok, showed that the problem admits an FPTAS on graphs of maximum degree $Δ$ as long as $γ$ is not a negative real number less than or equal to $-1/(4(Δ-1))$. Our first main result completes the picture for the approximability of the mat…
In order to understand the parameter space of monic and centered complex polynomial vector fields of degree d in the complex plane, decomposed by the combinatorial classes of the vector fields, it is interesting to know the number of loci in parameter space consisting of vector fields with the same combinatorial data (corresponding to topological classification with fixed separatrices at infinity). This paper answers questions posed by Adam L. Epstein and Tan Lei about the total number of combinatorial classes and the number of combinatorial classes corresponding to loci of a specific (real) d…
This paper gives an explicit formula for the Ehrhart quasi-polynomial of certain 2-dimensional polyhedra in terms of invariants of surface quotient singularities. Also, a formula for the dimension of the space of quasi-homogeneous polynomials of a given degree is derived. This admits an interpretation as a Numerical Adjunction Formula for singular curves on the weighted projective plane.
MacMahon's classic generating function of random plane partitions, which is related to Schur polynomials, was recently extended by Vuletic to a generating function of weighted plane partitions that is related to Hall-Littlewood polynomials, S(t), and further to one related to Macdonald polynomials, S(t,q). Using Jing's 1-parameter deformation of charged free fermions, we obtain a Fock space derivation of the Hall-Littlewood extension. Confining the plane partitions to a finite s-by-s square base, we show that the resulting generating function, S_{s-by-s}(t), is an evaluation of a tau-function …
We propose the concepts of intersection distribution and non-hitting index, which can be viewed from two related perspectives. The first one concerns a point set $S$ of size $q+1$ in the classical projective plane $PG(2,q)$, where the intersection distribution of $S$ indicates the intersection pattern between $S$ and the lines in $PG(2,q)$. The second one relates to a polynomial $f$ over a finite field $\mathbb{F}_q$, where the intersection distribution of $f$ records an overall distribution property of a collection of polynomials $\{f(x)+cx \mid c \in \mathbb{F}_q\}$. These two perspectives a…
We present multiresidue formulae for partial sums in the basis of link patterns of the polynomial solution to the level 1 U_q(\hat sl_2) quantum Knizhnik--Zamolodchikov equation at generic values of the quantum parameter q. These allow for rewriting and generalizing a recent conjecture [Di Francesco '06] connecting the above to generating polynomials for weighted Totally Symmetric Self-Complementary Plane Partitions. We reduce the corresponding conjectures to a single integral identity, yet to be proved.
