Liz McMahon - Profile on Academia.edu (original) (raw)
Papers by Liz McMahon
14. Error Detection and Correction Using SET
The Mathematics of Various Entertaining Subjects, 2016
Hokkaido Mathematical Journal, 1984
Advances in Applied Mathematics, 1997
Given a group G with generators ∆, it is well-known that the set of color-preserving automorphism... more Given a group G with generators ∆, it is well-known that the set of color-preserving automorphisms of the Cayley color digraph Γ = Cay ∆ (G) is isomorphic to G. Many people have studied the question of when the full automorphism group of the Cayley digraph is isomorphic to G. This paper explores what happens when the full automorphism group of G is not isomorphic to G: how much larger it can be and what kinds of structures can be found. The group of automorphisms that permute the color classes is a semidirect product; we look more closely at that and other sub-structures.
A greedoid characteristic polynomial
Contemporary Mathematics, 1996
Contemporary Mathematics Volume 197, 1996 A greedoid characteristic polynomial Gary Gordon and El... more Contemporary Mathematics Volume 197, 1996 A greedoid characteristic polynomial Gary Gordon and Elizabeth McMahon ABSTRACT. We define a characteristic polynomial p (G) for a greedoid G, gen-eralizing the well studied matroid characteristic polynomial (which in turn ...
Proceedings of the American Mathematical Society, 1989
We define a two-variable polynomial fa(t, z) for a greedoid G which generalizes the standard one-... more We define a two-variable polynomial fa(t, z) for a greedoid G which generalizes the standard one-variable greedoid polynomial A<j(f). Several greedoid invariants (including the number of feasible sets, bases, and spanning sets) are easily shown to be evaluations of fG(t, z). We prove (Theorem 2.8) that when G is a rooted directed arborescence, fo(t, z) completely determines the arborescence. We also show the polynomial is irreducible over Z[t, z] for arborescences with only one edge directed out of the distinguished vertex. When G is a matroid, fc(t, z) coincides with the Tutte polynomial. We also give an example to show Theorem 2.8 fails for full greedoids. This example also shows fa(t, z) does not distinguish rooted arborescences among the class of all greedoids.
PRIMUS, 2013
SET ® is a fun, fast-paced game that contains a surprising amount of mathematics. We will look in... more SET ® is a fun, fast-paced game that contains a surprising amount of mathematics. We will look in particular at hands-on activities in combinatorics and probability, finite geometry, and linear algebra for students at various levels. We also include a fun extension to the game that illustrates some of the power of thinking mathematically about the game.
Journal of Graph Theory, 1993
We examine some properties of the 2-variable greedoid polynomial f(G;t, z) when G is the branchin... more We examine some properties of the 2-variable greedoid polynomial f(G;t, z) when G is the branching greedoid associated to a rooted graph or a rooted directed graph. For rooted digraphs, we show a factoring property of f(G;t,z) determines whether or not the rooted digraph has a directed cycle.
Discrete Mathematics, 2003
A characteristic polynomial was recently deÿned for greedoids, generalizing the notion for matroi... more A characteristic polynomial was recently deÿned for greedoids, generalizing the notion for matroids. When chordal graphs are viewed as antimatroids by shelling of simplicial vertices, the greedoid characteristic polynomial gives additional information about those graphs. In particular, the characteristic polynomial for a chordal graph is an alternating clique generating function and is expressible in terms of the clique decomposition of the graph. From it, one obtains an expression for the number of blocks in the graph in terms of clique sizes.
Discrete Mathematics, 2001
We consider the one-variable characteristic polynomial p(G;) in two settings. When G is a rooted ... more We consider the one-variable characteristic polynomial p(G;) in two settings. When G is a rooted digraph, we show that this polynomial essentially counts the number of sinks in G. When G is a rooted graph, we give combinatorial interpretations of several coe cients and the degree of p(G;). In particular, |p(G; 0)| is the number of acyclic orientations of G, while the degree of p(G;) gives the size of the minimum tree cover (every edge of G is adjacent to some edge of T), and the leading coe cient gives the number of such covers. Finally, we consider the class of rooted fans in detail; here p(G;) shows cyclotomic behavior.
Journal of graph theory, 1993
We examine some properties of the 2-variable greedoid polynomial f(G;t, z) when G is the branchin... more We examine some properties of the 2-variable greedoid polynomial f(G;t, z) when G is the branching greedoid associated to a rooted graph or a rooted directed graph. For rooted digraphs, we show a factoring property of f(G;t,z) determines whether or not the rooted digraph has a directed cycle.
Discrete Mathematics, 2003
A characteristic polynomial was recently deÿned for greedoids, generalizing the notion for matroi... more A characteristic polynomial was recently deÿned for greedoids, generalizing the notion for matroids. When chordal graphs are viewed as antimatroids by shelling of simplicial vertices, the greedoid characteristic polynomial gives additional information about those graphs. In particular, the characteristic polynomial for a chordal graph is an alternating clique generating function and is expressible in terms of the clique decomposition of the graph. From it, one obtains an expression for the number of blocks in the graph in terms of clique sizes.
A greedoid characteristic polynomial
Matroid theory: AMS-IMS-SIAM Joint …, 1996
Contemporary Mathematics Volume 197, 1996 A greedoid characteristic polynomial Gary Gordon and El... more Contemporary Mathematics Volume 197, 1996 A greedoid characteristic polynomial Gary Gordon and Elizabeth McMahon ABSTRACT. We define a characteristic polynomial p (G) for a greedoid G, gen-eralizing the well studied matroid characteristic polynomial (which in turn ...
Proc. Amer. Math. Soc, 1989
Abstract. We define a two-variable polynomial fa(t, z) for a greedoid G which generalizes the sta... more Abstract. We define a two-variable polynomial fa(t, z) for a greedoid G which generalizes the standard one-variable greedoid polynomial A<j(f). Several greedoid invariants (including the number of feasible sets, bases, and spanning sets) are easily shown to be evaluations of ...
The American Mathematical Monthly, 2010
Derangements are a popular topic in combinatorics classes. We study a generalization to face dera... more Derangements are a popular topic in combinatorics classes. We study a generalization to face derangements of the n-dimensional hypercube. These derangements can be classified as odd or even, depending on whether the underlying isometry is direct or indirect, providing a link to abstract algebra. We emphasize the interplay between the geometry, algebra and combinatorics of these sequences, with lots of pretty pictures.
14. Error Detection and Correction Using SET
The Mathematics of Various Entertaining Subjects, 2016
Hokkaido Mathematical Journal, 1984
Advances in Applied Mathematics, 1997
Given a group G with generators ∆, it is well-known that the set of color-preserving automorphism... more Given a group G with generators ∆, it is well-known that the set of color-preserving automorphisms of the Cayley color digraph Γ = Cay ∆ (G) is isomorphic to G. Many people have studied the question of when the full automorphism group of the Cayley digraph is isomorphic to G. This paper explores what happens when the full automorphism group of G is not isomorphic to G: how much larger it can be and what kinds of structures can be found. The group of automorphisms that permute the color classes is a semidirect product; we look more closely at that and other sub-structures.
A greedoid characteristic polynomial
Contemporary Mathematics, 1996
Contemporary Mathematics Volume 197, 1996 A greedoid characteristic polynomial Gary Gordon and El... more Contemporary Mathematics Volume 197, 1996 A greedoid characteristic polynomial Gary Gordon and Elizabeth McMahon ABSTRACT. We define a characteristic polynomial p (G) for a greedoid G, gen-eralizing the well studied matroid characteristic polynomial (which in turn ...
Proceedings of the American Mathematical Society, 1989
We define a two-variable polynomial fa(t, z) for a greedoid G which generalizes the standard one-... more We define a two-variable polynomial fa(t, z) for a greedoid G which generalizes the standard one-variable greedoid polynomial A<j(f). Several greedoid invariants (including the number of feasible sets, bases, and spanning sets) are easily shown to be evaluations of fG(t, z). We prove (Theorem 2.8) that when G is a rooted directed arborescence, fo(t, z) completely determines the arborescence. We also show the polynomial is irreducible over Z[t, z] for arborescences with only one edge directed out of the distinguished vertex. When G is a matroid, fc(t, z) coincides with the Tutte polynomial. We also give an example to show Theorem 2.8 fails for full greedoids. This example also shows fa(t, z) does not distinguish rooted arborescences among the class of all greedoids.
PRIMUS, 2013
SET ® is a fun, fast-paced game that contains a surprising amount of mathematics. We will look in... more SET ® is a fun, fast-paced game that contains a surprising amount of mathematics. We will look in particular at hands-on activities in combinatorics and probability, finite geometry, and linear algebra for students at various levels. We also include a fun extension to the game that illustrates some of the power of thinking mathematically about the game.
Journal of Graph Theory, 1993
We examine some properties of the 2-variable greedoid polynomial f(G;t, z) when G is the branchin... more We examine some properties of the 2-variable greedoid polynomial f(G;t, z) when G is the branching greedoid associated to a rooted graph or a rooted directed graph. For rooted digraphs, we show a factoring property of f(G;t,z) determines whether or not the rooted digraph has a directed cycle.
Discrete Mathematics, 2003
A characteristic polynomial was recently deÿned for greedoids, generalizing the notion for matroi... more A characteristic polynomial was recently deÿned for greedoids, generalizing the notion for matroids. When chordal graphs are viewed as antimatroids by shelling of simplicial vertices, the greedoid characteristic polynomial gives additional information about those graphs. In particular, the characteristic polynomial for a chordal graph is an alternating clique generating function and is expressible in terms of the clique decomposition of the graph. From it, one obtains an expression for the number of blocks in the graph in terms of clique sizes.
Discrete Mathematics, 2001
We consider the one-variable characteristic polynomial p(G;) in two settings. When G is a rooted ... more We consider the one-variable characteristic polynomial p(G;) in two settings. When G is a rooted digraph, we show that this polynomial essentially counts the number of sinks in G. When G is a rooted graph, we give combinatorial interpretations of several coe cients and the degree of p(G;). In particular, |p(G; 0)| is the number of acyclic orientations of G, while the degree of p(G;) gives the size of the minimum tree cover (every edge of G is adjacent to some edge of T), and the leading coe cient gives the number of such covers. Finally, we consider the class of rooted fans in detail; here p(G;) shows cyclotomic behavior.
Journal of graph theory, 1993
We examine some properties of the 2-variable greedoid polynomial f(G;t, z) when G is the branchin... more We examine some properties of the 2-variable greedoid polynomial f(G;t, z) when G is the branching greedoid associated to a rooted graph or a rooted directed graph. For rooted digraphs, we show a factoring property of f(G;t,z) determines whether or not the rooted digraph has a directed cycle.
Discrete Mathematics, 2003
A characteristic polynomial was recently deÿned for greedoids, generalizing the notion for matroi... more A characteristic polynomial was recently deÿned for greedoids, generalizing the notion for matroids. When chordal graphs are viewed as antimatroids by shelling of simplicial vertices, the greedoid characteristic polynomial gives additional information about those graphs. In particular, the characteristic polynomial for a chordal graph is an alternating clique generating function and is expressible in terms of the clique decomposition of the graph. From it, one obtains an expression for the number of blocks in the graph in terms of clique sizes.
A greedoid characteristic polynomial
Matroid theory: AMS-IMS-SIAM Joint …, 1996
Contemporary Mathematics Volume 197, 1996 A greedoid characteristic polynomial Gary Gordon and El... more Contemporary Mathematics Volume 197, 1996 A greedoid characteristic polynomial Gary Gordon and Elizabeth McMahon ABSTRACT. We define a characteristic polynomial p (G) for a greedoid G, gen-eralizing the well studied matroid characteristic polynomial (which in turn ...
Proc. Amer. Math. Soc, 1989
Abstract. We define a two-variable polynomial fa(t, z) for a greedoid G which generalizes the sta... more Abstract. We define a two-variable polynomial fa(t, z) for a greedoid G which generalizes the standard one-variable greedoid polynomial A<j(f). Several greedoid invariants (including the number of feasible sets, bases, and spanning sets) are easily shown to be evaluations of ...
The American Mathematical Monthly, 2010
Derangements are a popular topic in combinatorics classes. We study a generalization to face dera... more Derangements are a popular topic in combinatorics classes. We study a generalization to face derangements of the n-dimensional hypercube. These derangements can be classified as odd or even, depending on whether the underlying isometry is direct or indirect, providing a link to abstract algebra. We emphasize the interplay between the geometry, algebra and combinatorics of these sequences, with lots of pretty pictures.