26. CMB 2010 (vol 53 pp. 757)
 Woo, Alexander

Interval Pattern Avoidance for Arbitrary Root Systems
We extend the idea of interval pattern avoidance defined by Yong and
the author for $S_n$ to arbitrary Weyl groups using the definition of
pattern avoidance due to Billey and Braden, and Billey and Postnikov.
We show that, as previously shown by Yong and the
author for $\operatorname{GL}_n$, interval pattern avoidance is a universal tool for
characterizing which Schubert varieties have certain local properties,
and where these local properties hold.
Categories:14M15, 05E15 

27. CMB 2010 (vol 53 pp. 425)
 Chapoton, Frédéric

Free PreLie Algebras are Free as Lie Algebras
We prove that the $\mathfrak{S}$module $\operatorname{PreLie}$ is a free Lie algebra in
the category of $\mathfrak{S}$modules and can therefore be written as the
composition of the $\mathfrak{S}$module $\operatorname{Lie}$ with a new $\mathfrak{S}$module
$X$. This implies that free preLie algebras in the category of
vector spaces, when considered as Lie algebras, are free on
generators that can be described using $X$. Furthermore, we define a
natural filtration on the $\mathfrak{S}$module $X$. We also obtain a
relationship between $X$ and the $\mathfrak{S}$module coming from the
anticyclic structure of the $\operatorname{PreLie}$ operad.
Categories:18D50, 17B01, 18G40, 05C05 

28. CMB 2010 (vol 53 pp. 453)
29. CMB 2009 (vol 53 pp. 378)
 Zhou, Sizhong

A New Sufficient Condition for a Graph To Be $(g,f,n)$Critical
Let $G$ be a graph of order $p$, let $a$,
$b$, and $n$ be nonnegative integers with $1\leq a\lt b$, and let $g$
and $f$ be two integervalued functions defined on $V(G)$ such
that $a\leq g(x)\lt f(x)\leq b$ for all $x\in V(G)$. A $(g,f)$factor
of graph $G$ is a spanning subgraph $F$ of $G$ such
that $g(x)\leq d_F(x)\leq f(x)$ for each $x\in V(F)$. Then a graph
$G$ is called $(g,f,n)$critical if after deleting any $n$
vertices of $G$ the remaining graph of $G$ has a $(g,f)$factor.
The binding number $\operatorname{bind}(G)$ of $G$ is the minimum value of
${N_G(X)}/{X}$ taken over all nonempty subsets $X$ of
$V(G)$ such that $N_G(X)\neq V(G)$. In this paper, it is proved
that $G$ is a $(g,f,n)$critical graph if
\[
\operatorname{bind}(G)\gt \frac{(a+b1)(p1)}{(a+1)p(a+b)bn+2}
\quad\text{and}\quad p\geq
\frac{(a+b1)(a+b2)}{a+1}+\frac{bn}{a}.
\]
Furthermore, it is
shown that this
result is best possible in some sense.
Keywords:graph, $(g,f)$factor, $(g,f,n)$critical graph, binding number Category:05C70 

30. CMB 2009 (vol 53 pp. 3)
 Athanasiadis, Christos A.

A Combinatorial Reciprocity Theorem for Hyperplane Arrangements
Given a nonnegative integer $m$ and a finite collection $\mathcal A$ of
linear forms on $\mathcal Q^d$, the arrangement of affine hyperplanes in
$\mathcal Q^d$ defined by the equations $\alpha(x) = k$ for $\alpha
\in \mathcal A$
and integers $k \in [m, m]$ is denoted by $\mathcal A^m$. It is proved that
the coefficients of the characteristic polynomial of $\mathcal A^m$ are
quasipolynomials in $m$ and that they satisfy a simple combinatorial
reciprocity law.
Categories:52C35, 05E99 

31. CMB 2009 (vol 53 pp. 171)
 Thomas, Hugh; Yong, Alexander

MultiplicityFree Schubert Calculus
Multiplicityfree algebraic geometry is the study of subvarieties
$Y\subseteq X$ with the ``smallest invariants'' as witnessed by a
multiplicityfree Chow ring decomposition of
$[Y]\in A^{\star}(X)$ into a predetermined
linear basis.
This paper concerns the case of Richardson subvarieties of the Grassmannian
in terms of the Schubert basis. We give a nonrecursive combinatorial
classification of multiplicityfree Richardson varieties, i.e.,
we classify multiplicityfree products of Schubert classes. This answers
a question of W. Fulton.
Categories:14M15, 14M05, 05E99 

32. CMB 2009 (vol 52 pp. 416)
 Malik, Shabnam; Qureshi, Ahmad Mahmood; Zamfirescu, Tudor

Hamiltonian Properties of Generalized Halin Graphs
A Halin graph is a graph $H=T\cup C$, where $T$ is a tree with no
vertex of degree two, and $C$ is a cycle connecting the endvertices
of $T$ in the cyclic order determined by a plane embedding of $T$.
In this paper, we define classes of generalized Halin graphs, called
$k$Halin graphs, and investigate their Hamiltonian properties.
Keywords:$k$Halin graph, Hamiltonian, Hamiltonian connected, traceable Categories:05C45, 05C38 

33. CMB 2009 (vol 52 pp. 451)
 Pach, János; Tardos, Gábor; Tóth, Géza

Indecomposable Coverings
We prove that for every $k>1$, there exist $k$fold coverings of the
plane (i) with strips, (ii) with axisparallel rectangles, and
(iii) with homothets of any fixed concave quadrilateral, that cannot
be decomposed into two coverings. We also construct for every
$k>1$ a set of points $P$ and a family of disks $\cal D$ in the
plane, each containing at least $k$ elements of $P$, such that, no
matter how we color the points of $P$ with two colors,
there
exists a disk $D\in{\cal D}$ all of whose points are of the same
color.
Categories:52C15, 05C15 

34. CMB 2008 (vol 51 pp. 584)
 Purbhoo, Kevin; Willigenburg, Stephanie van

On Tensor Products of Polynomial Representations
We determine the necessary and sufficient combinatorial
conditions for which the tensor product of two irreducible polynomial
representations of $\GL(n,\mathbb{C})$ is isomorphic to another.
As a consequence we discover families of LittlewoodRichardson
coefficients that are nonzero, and a condition on Schur nonnegativity.
Keywords:polynomial representation, symmetric function, LittlewoodRichardson coefficient, Schur nonnegative Categories:05E05, 05E10, 20C30 

35. CMB 2008 (vol 51 pp. 535)
36. CMB 2008 (vol 51 pp. 413)
37. CMB 2008 (vol 51 pp. 424)
38. CMB 2008 (vol 51 pp. 47)
 Croot, Ernie

The Minimal Number of ThreeTerm Arithmetic Progressions Modulo a Prime Converges to a Limit
How few threeterm arithmetic progressions can a
subset $S \subseteq \Z_N := \Z/N\Z$ have if $S \geq \upsilon N$
(that is, $S$ has density at least $\upsilon$)?
Varnavides %\cite{varnavides}
showed that this number of arithmetic progressions is at
least $c(\upsilon)N^2$ for sufficiently large integers $N$.
It is well known that determining good lower bounds for
$c(\upsilon)> 0$ is at the same level of depth as Erd\" os's famous
conjecture about whether a subset $T$ of the naturals where
$\sum_{n \in T} 1/n$ diverges, has a $k$term arithmetic progression
for $k=3$ (that is, a threeterm arithmetic progression).
We answer a question posed by B. Green %\cite{AIM}
about how this minimial number of progressions oscillates
for a fixed density $\upsilon$ as $N$ runs through the primes, and
as $N$ runs through the odd positive integers.
Category:05D99 

39. CMB 2007 (vol 50 pp. 504)
 Dukes, Peter; Ling, Alan C. H.

Asymptotic Existence of Resolvable Graph Designs
Let $v \ge k \ge 1$ and $\lam \ge 0$ be integers. A \emph{block
design} $\BD(v,k,\lambda)$ is a collection $\cA$ of $k$subsets of a
$v$set $X$ in which every unordered pair of elements from $X$ is
contained in exactly $\lambda$ elements of $\cA$. More generally, for a
fixed simple graph $G$, a \emph{graph design} $\GD(v,G,\lambda)$ is a
collection $\cA$ of graphs isomorphic to $G$ with vertices in $X$ such
that every unordered pair of elements from $X$ is an edge of exactly
$\lambda$ elements of $\cA$. A famous result of Wilson says that for a
fixed $G$ and $\lambda$, there exists a $\GD(v,G,\lambda)$ for all
sufficiently large $v$ satisfying certain necessary conditions. A
block (graph) design as above is \emph{resolvable} if $\cA$ can be
partitioned into partitions of (graphs whose vertex sets partition)
$X$. Lu has shown asymptotic existence in $v$ of resolvable
$\BD(v,k,\lambda)$, yet for over twenty years the analogous problem for
resolvable $\GD(v,G,\lambda)$ has remained open. In this paper, we settle
asymptotic existence of resolvable graph designs.
Keywords:graph decomposition, resolvable designs Categories:05B05, 05C70, 05B10 

40. CMB 2007 (vol 50 pp. 632)
 Zelenyuk, Yevhen; Zelenyuk, Yuliya

Transformations and Colorings of Groups
Let $G$ be a compact topological group and let $f\colon G\to G$ be a
continuous transformation of $G$. Define $f^*\colon G\to G$ by
$f^*(x)=f(x^{1})x$ and let $\mu=\mu_G$ be Haar measure on $G$. Assume
that $H=\Imag f^*$ is a subgroup of $G$ and for every
measurable $C\subseteq H$,
$\mu_G((f^*)^{1}(C))=\mu_H(C)$. Then for every measurable
$C\subseteq G$, there exist $S\subseteq C$ and $g\in G$ such that
$f(Sg^{1})\subseteq Cg^{1}$ and $\mu(S)\ge(\mu(C))^2$.
Keywords:compact topological group, continuous transformation, endomorphism, Ramsey theoryinversion, Categories:05D10, 20D60, 22A10 

41. CMB 2007 (vol 50 pp. 535)
 Hohlweg, Christophe

Generalized Descent Algebras
If $A$ is a subset of the set of reflections of a finite Coxeter
group $W$, we define a sub$\ZM$module $\DC_A(W)$ of the group
algebra $\ZM W$. We discuss cases where this submodule is a
subalgebra. This family of subalgebras includes strictly the
Solomon descent algebra, the group algebra and, if $W$ is of type
$B$, the MantaciReutenauer algebra.
Keywords:Coxeter group, Solomon descent algebra, descent set Categories:20F55, 05E15 

42. CMB 2006 (vol 49 pp. 281)
 Ragnarsson, Carl Johan; Suen, Wesley Wai; Wagner, David G.

Correction to a Theorem on Total Positivity
A wellknown theorem states that if $f(z)$ generates a PF$_r$
sequence then $1/f(z)$ generates a PF$_r$ sequence. We give two
counterexamples
which show that this is not true, and give a correct version of the theorem.
In the infinite limit the result is sound: if $f(z)$ generates a PF
sequence then $1/f(z)$ generates a PF sequence.
Keywords:total positivity, Toeplitz matrix, PÃ³lya frequency sequence, skew Schur function Categories:15A48, 15A45, 15A57, 05E05 

43. CMB 2005 (vol 48 pp. 445)
 Patras, Frédéric; Reutenauer, Christophe; Schocker, Manfred

On the Garsia Lie Idempotent
The orthogonal projection of the free associative algebra onto the
free Lie algebra is afforded by an idempotent in the rational group
algebra of the symmetric group $S_n$, in each homogenous degree
$n$. We give various characterizations of this Lie idempotent and show
that it is uniquely determined by a certain unit in the group algebra
of $S_{n1}$. The inverse of this unit, or, equivalently, the Gram
matrix of the orthogonal projection, is described explicitly. We also
show that the Garsia Lie idempotent is not constant on descent classes
(in fact, not even on coplactic classes) in $S_n$.
Categories:17B01, 05A99, 16S30, 17B60 

44. CMB 2005 (vol 48 pp. 460)
 Sommers, Eric N.

$B$Stable Ideals in the Nilradical of a Borel Subalgebra
We count the number of strictly positive $B$stable ideals in the
nilradical of a Borel subalgebra and prove that
the minimal roots of any $B$stable ideal are conjugate
by an element of the Weyl group to a subset of the simple roots.
We also count the number of ideals whose minimal roots are conjugate
to a fixed subset of simple roots.
Categories:20F55, 17B20, 05E99 

45. CMB 2005 (vol 48 pp. 244)
 McLeod, Alice; Moser, William

Counting Multiple Cyclic Choices Without Adjacencies
We give a particularly elementary solution to the following
wellknown problem. What is the number of $k$subsets $X \subseteq
I_n=\{1,2,3,\dots,n\}$ satisfying ``no two elements of $X$ are adjacent
in the circular display of $I_n$''? Then we investigate a new
generalization (multiple cyclic choices without adjacencies) and
apply it to enumerating a class of 3line latin rectangles.
Categories:05A19, 05A05 

46. CMB 2004 (vol 47 pp. 161)
47. CMB 2002 (vol 45 pp. 537)
48. CMB 2002 (vol 45 pp. 321)
 Brenken, Berndt

$C^{\ast}$Algebras of Infinite Graphs and CuntzKrieger Algebras
The CuntzKrieger algebra $\mathcal{O}_B$ is defined for an
arbitrary, possibly infinite and infinite valued, matrix $B$. A graph
$C^{\ast}$algebra $G^{\ast} (E)$ is introduced for an arbitrary
directed graph $E$, and is shown to coincide with a previously defined
graph algebra $C^{\ast} (E)$ if each source of $E$ emits only finitely
many edges. Each graph algebra $G^{\ast} (E)$ is isomorphic to the
CuntzKrieger algebra $\mathcal{O}_B$ where $B$ is the vertex matrix
of~$E$.
Categories:46LXX, 05C50 

49. CMB 2001 (vol 44 pp. 370)
 Weston, Anthony

On Locating Isometric $\ell_{1}^{(n)}$
Motivated by a question of Per Enflo, we develop a hypercube criterion
for locating linear isometric copies of $\lone$ in an arbitrary real
normed space $X$.
The said criterion involves finding $2^{n}$ points in $X$ that satisfy
one metric equality. This contrasts nicely to the standard classical
criterion wherein one seeks $n$ points that satisfy $2^{n1}$ metric
equalities.
Keywords:normed spaces, hypercubes Categories:46B04, 05C10, 05B99 

50. CMB 2000 (vol 43 pp. 397)