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"Brualdi, Richard A"
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Multipermutations and Stirling Multipermutations
We consider multipermutations and a certain partial order, the weak Bruhat order, on this set. This generalizes the Bruhat order for permutations, and is defined in terms of containment of inversions. Different characterizations of this order are given. We also study special multipermutations called Stirling multipermutations and their properties.
Journal Article
Completions of Alternating Sign Matrices
We consider the problem of completing a
(
0
,
-
1
)
-matrix to an alternating sign matrix (ASM) by replacing some
0
s with
-
1
s. An algorithm can be given to determine a completion or show that one does not exist. We are concerned primarily with bordered-permutation
(
0
,
-
1
)
matrices, defined to be
n
×
n
(
0
,
-
1
)
-matrices with only
0
s in their first and last rows and columns where the
-
1
s form an
(
n
-
2
)
×
(
n
-
2
)
permutation matrix. We show that any such matrix can be completed to an ASM and characterize those that have a unique completion.
Journal Article
Nested Species Subsets, Gaps, and Discrepancy
The nested-subset hypothesis of Patterson and Atmar states that species composition on islands with less species richness is a proper subset of those on islands with greater species richness. The sum of species absences, referred to as gaps, was suggested as a metric for nestedness, and null models have been used to test whether or not island species exhibited nestedness. Simberloff and Martin stated that finding examples of non-nested faunas was difficult. We revisit previous analyses of nested faunas and introduce a new metric we call \"discrepancy\" which we recommend as a measure for nestedness. We also recommend that the sample spaces conserve both row sums (number of species per site) and column sums (number of sites per species) derived from the incidence matrix. We compare our results to previous analyses.
Journal Article
Sum List Coloring Graphs
Let G=(V,E) be a graph with n vertices and e edges. The sum choice number of G is the smallest integer p such that there exist list sizes (f(v):v belongs to V) whose sum is p for which G has a proper coloring no matter which color lists of size f(v) are assigned to the vertices v. The sum choice number is bounded above by n+e. If the sum choice number of G equals n+e, then G is sum choice greedy. Complete graphs K[subscript n] are sum choice greedy as are trees. Based on a simple, but powerful, lemma we show that a graph each of whose blocks is sum choice greedy is also sum choice greedy. We also determine the sum choice number of K[subscript 2,n], and we show that every tree on n vertices can be obtained from K[subscript n] by consecutively deleting single edges where all intermediate graphs are sc-greedy. [PUBLICATION ABSTRACT]
Journal Article
Combinatorics and graphs: Twentieth Anniversary Conference of IPM Combinatorics, Tehran, Iran
This volume contains a collection of papers presented at the international conference IPM 20--Combinatorics 2009, which was held at the Institute for Research in Fundamental Sciences in Tehran, Iran, May 15-21, 2009. The conference celebrated IPM's 20th anniversary and was dedicated to Reza Khosrovshahi, one of the founders of IPM and the director of its School of Mathematics from 1996 to 2007, on the occasion of his 70th birthday. The conference attracted an international group of distinguished researchers from many different parts of combinatorics and graph theory, including permutations, designs, graph minors, graph coloring, graph eigenvalues, distance regular graphs and association schemes, hypergraphs, and arrangements.|This volume contains a collection of papers presented at the international conference IPM 20--Combinatorics 2009, which was held at the Institute for Research in Fundamental Sciences in Tehran, Iran, May 15-21, 2009. The conference celebrated IPM's 20th anniversary and was dedicated to Reza Khosrovshahi, one of the founders of IPM and the director of its School of Mathematics from 1996 to 2007, on the occasion of his 70th birthday. The conference attracted an international group of distinguished researchers from many different parts of combinatorics and graph theory, including permutations, designs, graph minors, graph coloring, graph eigenvalues, distance regular graphs and association schemes, hypergraphs, and arrangements.
eBook
The Sparse Basis Problem and Multilinear Algebra
Let $A$ be a $k \\times n$ underdetermined matrix. The sparse basis problem for the row space $W$ of $A$ is to find a basis of $W$ with the fewest number of nonzeros. Suppose that all the entries of $A$ are nonzero, and that they are algebraically independent over the rational number field. Then every nonzero vector in $W$ has at least $n - k + 1$ nonzero entries. Those vectors in $W$ with exactly $n - k + 1$ nonzero entries are the elementary vectors of $W$. A simple combinatorial condition that is both necessary and sufficient for a set of $k$ elementary vectors of $W$ to form a basis of $W$ is presented here. A similar result holds for the null space of $A$ where the elementary vectors now have exactly $k + 1$ nonzero entries. These results follow from a theorem about nonzero minors of order $m$ of the $(m - 1)$st compound of an $m \\times n$ matrix with algebraically independent entries, which is proved using multilinear algebra techniques. This combinatorial condition for linear independence is a first step towards the design of algorithms that compute sparse bases for the row and null space without imposing artificial structure constraints to ensure linear independence.
Journal Article