51. CMB 2010 (vol 53 pp. 453)
52. 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 

53. 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 

54. 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 

55. 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 

56. 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 

57. CMB 2009 (vol 40 pp. 149)
 Brown, Tom C.; Landman, Bruce M.; Mishna, Marni

Monochromatic homothetic copies\\ of $\{1,1+s,1+s+t\}$
For positive integers $s$ and $t$, let $f(s, t)$ denote the smallest positive
integer $N$ such that every $2$colouring of $[1,N]=\{1,2, \ldots , N\}$ has
a monochromatic homothetic copy of $\{1, 1+s, 1+s+t\}$.
We show that $f(s, t) = 4(s+t) + 1$ whenever $s/g$ and $t/g$ are not
congruent to $0$ (modulo $4$), where $g=\gcd(s,t)$. This can be viewed as
a generalization of part of van~der~Waerden's theorem on
arithmetic progressions, since the $3$term arithmetic progressions are the
homothetic copies of $\{1, 1+1, 1+1+1\}$. We also show that $f(s, t) = 4(s+t)
+ 1$ in many other cases (for example, whenever $s > 2t > 2$ and $t$ does not
divide $s$), and that $f(s, t) \le 4(s+t) + 1$ for all $s$, $t$.
Thus the set of homothetic copies of $\{1, 1+s, 1+s+t\}$ is a set of
triples with a particularly simple Ramsey function (at least for the case
of two colours), and one wonders what other ``natural'' sets of triples,
quadruples, {\it etc.}, have simple (or easily estimated) Ramsey functions.
Category:05D10 

58. 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 

59. CMB 2008 (vol 51 pp. 535)
60. CMB 2008 (vol 51 pp. 424)
61. CMB 2008 (vol 51 pp. 413)
62. 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 

63. 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 

64. 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 

65. 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 

66. 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 

67. 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 

68. 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 

69. 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 

70. CMB 2004 (vol 47 pp. 161)
71. CMB 2002 (vol 45 pp. 537)
72. 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 

73. 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 

74. CMB 2000 (vol 43 pp. 397)
75. CMB 2000 (vol 43 pp. 385)
 Bluskov, I.; Greig, M.; Heinrich, K.

Infinite Classes of Covering Numbers
Let $D$ be a family of $k$subsets (called blocks) of a $v$set
$X(v)$. Then $D$ is a $(v,k,t)$ covering design or covering if every
$t$subset of $X(v)$ is contained in at least one block of $D$. The
number of blocks is the size of the covering, and the minimum size of
the covering is called the covering number. In this paper we consider
the case $t=2$, and find several infinite classes of covering numbers.
We also give upper bounds on other classes of covering numbers.
Categories:05B40, 05D05 
