An ordered set-partition (or preferential arrangement) of n labeled elements represents a single ``hierarchy''; these are enumerated by the ordered Bell numbers. In this note we determine the number of ``hierarchical orderings'' or ``societies'', where the n elements are first partitioned into m <= n subsets and a hierarchy is specified for each subset. We also consider the unlabeled case, where the ordered Bell numbers are replaced by the composition numbers. If there is only a single hierarchy, we show that the average rank of an element is asymptotic to n/(4 log 2) in the labeled case and to n/4 in the unlabeled case.
The Stirling numbers of the second kind are related to normal orderings in the Weyl algebra, while the unsigned Stirling numbers of the first kind are related to normal orderings in the shift algebra. Kim-Kim introduced a λ-analogue of the unsigned Stirling numbers of the first kind and that of the r-Stirling numbers of the first kind. In this paper, we introduce a λ-analogue of the shift algebra (called λ-shift algebra) and investigate normal orderings in the λ-shift algebra. From the normal orderings in the λ-shift algebra, we derive some identities about the λ-analogue of the unsigned Stirl…
A $k$-ordering of a graph $G$ assigns distinct order-labels from the set $\{1,\ldots,|G|\}$ to $k$ vertices in $G$. Given a $k$-ordering $H$, the ordered Ramsey number $R_<(H)$ is the minimum $n$ such that every edge-2-coloring of the complete graph on the vertex set $\{1, \ldots, n\}$ contains a copy of $H$, the $i$th smallest vertex of which either has order-label $i$ in $H$ or no order-label in $H$. This paper conducts the first systematic study of ordered Ramsey numbers for $1$-orderings of small graphs. We provide upper bounds for $R_<(H)$ for each connected $1$-ordering $H$ on $4$ vertic…
The Stirling number of a simple graph is the number of partitions of its vertex set into a specific number of non-empty independent sets. In 2015, Engbers et al. showed that the coefficients in the normal ordering of a word $w$ in the alphabet $\{x,D\}$ subject to the relation $Dx=xD+1$ are equal to the Stirling number of certain graphs constructed from $w$. In this paper, we introduce graphical versions of the Stirling numbers of the first kind and the Lah numbers and show how they occur as coefficients in other normal ordering settings. Identities involving their $q$-analogues are also obtai…
Given a set $S$ of $n$ points in the plane, a \emph{radial ordering} of $S$ with respect to a point $p$ (not in $S$) is a clockwise circular ordering of the elements in $S$ by angle around $p$. If $S$ is two-colored, a \emph{colored radial ordering} is a radial ordering of $S$ in which only the colors of the points are considered. In this paper, we obtain bounds on the number of distinct non-colored and colored radial orderings of $S$. We assume a strong general position on $S$, not three points are collinear and not three lines---each passing through a pair of points in $S$---intersect in a p…
We discuss the validity of the proof of the fixed numerator conjecture on Markov numbers, which is the main result of the paper mentioned in the title.
Balogh, Bollobas and Weinreich showed that a parameter that has since been termed the distinguishing number can be used to identify a jump in the possible speeds of hereditary classes of graphs at the sequence of Bell numbers. We prove that every hereditary class that lies above the Bell numbers and has finite distinguishing number contains a boundary class for well-quasi-ordering. This means that any such hereditary class which in addition is defined by finitely many minimal forbidden induced subgraphs must contain an infinite antichain. As all hereditary classes below the Bell numbers are we…
The generalized coloring numbers col_r(G) (also denoted by scol_r(G)) and wcol_r(G) of a graph G were introduced by Kierstead and Yang as a generalization of the usual coloring number, and have found important theoretical and algorithmic applications. For each distance r, these numbers are determined by an "optimal" ordering of the vertices of G. We study the question of whether it is possible to find a single "uniform" ordering that is "good" for all distances r. We show that the answer to this question is essentially "yes". Our results give new characterizations of graph classes with bounded…
We consider the numbers arising in the problem of normal ordering of expressions in canonical boson creation and annihilation operators. We treat a general form of a boson string which is shown to be associated with generalizations of Stirling and Bell numbers. The recurrence relations and closed-form expressions (Dobiski-type formulas) are obtained for these quantities by both algebraic and combinatorial methods. By extensive use of methods of combinatorial analysis we prove the equivalence of the aforementioned problem to the enumeration of special families of graphs. This link provides a co…
The normal ordering coefficients of strings consisting of $V,U$ which satisfy $UV=qVU+hV^s$ ($s\in\mathbb N$) are considered. These coefficients are studied in two contexts: first, as a multiple of a sequence satisfying a generalized recurrence, and second, as $q$-analogues of rook numbers under the row creation rule introduced by Goldman and Haglund. A number of properties are derived, including recurrences, expressions involving other $q$-analogues and explicit formulas. We also give a Dobinsky-type formula for the associated Bell numbers and the corresponding extension of Spivey's Bell numb…
Markov numbers are integers that appear in the solution triples of the Diophantine equation, $x^2+y^2+z^2=3xyz$, called the Markov equation. A classical topic in number theory, these numbers are related to many areas of mathematics such as combinatorics, hyperbolic geometry, approximation theory and cluster algebras. There is a natural map from the rational numbers between zero and one to the Markov numbers. In this paper, we prove two conjectures seen in Martin Aigner's book, Markov's theorem and 100 years of the uniqueness conjecture, that determine an ordering on subsets of the Markov numbe…
A graph G = (V,E) is called fully regular if for every independent set $I\subset V$ , the number of vertices in $V\setminus$ I that are not connected to any element of I depends only on the size of I. A linear ordering of the vertices of G is called successive if for every i, the first i vertices induce a connected subgraph of G. We give an explicit formula for the number of successive vertex orderings of a fully regular graph. As an application of our results, we give alternative proofs of two theorems of Stanley and Gao + Peng, determining the number of linear edge orderings of complete grap…
For any function F(x) having a Taylor expansion we solve the boson normal ordering problem for F[(a*)^r a^s], with r,s positive integers,[a,a*]=1, i.e. we provide exact and explicit expressions for its normal form which has all a's to the right. The solution involves integer sequences of numbers which, for r,s >=1, are generalizations of the conventional Bell and Stirling numbers whose values they assume for r=s=1. A complete theory of such generalized combinatorial numbers is given including closed-form expressions (extended Dobinski - type formulas), recursion relations and generating functi…
For an element $w$ in the Weyl algebra generated by $D$ and $U$ with relation $DU=UD+1$, the normally ordered form is $w=\sum c_{i,j}U^iD^j$. We demonstrate that the normal order coefficients $c_{i,j}$ of a word $w$ are rook numbers on a Ferrers board. We use this interpretation to give a new proof of the rook factorization theorem, which we use to provide an explicit formula for the coefficients $c_{i,j}$. We calculate the Weyl binomial coefficients: normal order coefficients of the element $(D+U)^n$ in the Weyl algebra. We extend all these results to the $q$-analogue of the Weyl algebra. We …
We derive explicit formulas for the normal ordering of powers of arbitrary monomials of boson operators. These formulas lead to generalisations of conventional Bell and Stirling numbers and to appropriate generalisations of the Dobinski relations. These new combinatorial numbers are shown to be coherent state matrix elements of powers of the monomials in question. It is further demonstrated that such Bell-type numbers, when considered as power moments, give rise to positive measures on the positive half-axis, which in many cases can be written in terms of known functions.
We study Koebe orderings of planar graphs: vertex orderings obtained by modelling the graph as the intersection graph of pairwise internally-disjoint discs in the plane, and ordering the vertices by non-increasing radii of the associated discs. We prove that for every $d\in \mathbb{N}$, any such ordering has $d$-admissibility bounded by $O(d/\ln d)$ and weak $d$-coloring number bounded by $O(d^4 \ln d)$. This in particular shows that the $d$-admissibility of planar graphs is bounded by $O(d/\ln d)$, which asymptotically matches a known lower bound due to Dvořák and Siebertz.
We discuss a general combinatorial framework for operator ordering problems by applying it to the normal ordering of the powers and exponential of the boson number operator. The solution of the problem is given in terms of Bell and Stirling numbers enumerating partitions of a set. This framework reveals several inherent relations between ordering problems and combinatorial objects, and displays the analytical background to Wick's theorem. The methodology can be straightforwardly generalized from the simple example given herein to a wide class of operators.
We solve the boson normal ordering problem for F[(a*)^r a^s], with r,s positive integers, where a* and a are boson creation and annihilation operators satisfying [a,a*]=1. That is, we provide exact and explicit expressions for the normal form wherein all a's are to the right. The solution involves integer sequences of numbers which are generalizations of the conventional Bell and Stirling numbers whose values they assume for r=s=1. A comprehensive theory of such generalized combinatorial numbers is given including closed-form expressions (extended Dobinski-type formulas)and generating function…
An ordered graph $\mathcal{G}$ is a simple graph together with a total ordering on its vertices. The (2-color) Ramsey number of $\mathcal{G}$ is the smallest integer $N$ such that every 2-coloring of the edges of the complete ordered graph on $N$ vertices has a monochromatic copy of $\mathcal{G}$ that respects the ordering. In this paper we investigate the effect of various graph operations on the Ramsey number of a given ordered graph, and detail a general framework for applying results on extremal functions of 0-1 matrices to ordered Ramsey problems. We apply this method to give upper bounds…
The $k$-th spectral moment $M_k(G)$ of the adjacency matrix of a graph~$G$ represents the number of closed walks of length~$k$ in~$G$. We study here the partial order $\preceq$ of graphs, defined by $G\preceq H$ if $M_k(G)\leq M_k(H)$ for all $k\geq 0$, and are interested in the question when is $\preceq$ a linear order within a specified set of graphs? Our main result is that $\preceq$ is a linear order on each set of starlike trees with constant number of vertices. Recall that a connected graph $G$ is a starlike tree if it has a vertex~$u$ such that the components of $G-u$ are paths, called …
Dominator coloring of a graph is a proper (vertex) coloring with the property that every vertex is either alone in its color class or adjacent to all vertices of at least one color class. A dominated coloring of a graph is a proper coloring such that every color class is dominated with at least one vertex. The dominator chromatic number of corona products and of edge corona products is determined. Sharp lower and upper bounds are given for the dominated chromatic number of edge corona products. The dominator chromatic number of hierarchical products is bounded from above and the dominated chro…
The Markov numbers are the positive integers that appear in the solutions of the equation $x^2+y^2+z^2=3xyz$. These numbers are a classical subject in number theory and have important ramifications in hyperbolic geometry, algebraic geometry and combinatorics. It is known that the Markov numbers can be labeled by the lattice points $(q,p)$ in the first quadrant and below the diagonal whose coordinates are coprime. In this paper, we consider the following question. Given two lattice points, can we say which of the associated Markov numbers is larger? A complete answer to this question would solv…
