263 research outputs found
Enumerative properties of Ferrers graphs
We define a class of bipartite graphs that correspond naturally with Ferrers
diagrams. We give expressions for the number of spanning trees, the number of
Hamiltonian paths when applicable, the chromatic polynomial, and the chromatic
symmetric function. We show that the linear coefficient of the chromatic
polynomial is given by the excedance set statistic.Comment: 12 page
Enumerative Coding for Grassmannian Space
The Grassmannian space \Gr is the set of all dimensional subspaces of
the vector space~\smash{\F_q^n}. Recently, codes in the Grassmannian have
found an application in network coding. The main goal of this paper is to
present efficient enumerative encoding and decoding techniques for the
Grassmannian. These coding techniques are based on two different orders for the
Grassmannian induced by different representations of -dimensional subspaces
of \F_q^n. One enumerative coding method is based on a Ferrers diagram
representation and on an order for \Gr based on this representation. The
complexity of this enumerative coding is digit
operations. Another order of the Grassmannian is based on a combination of an
identifying vector and a reduced row echelon form representation of subspaces.
The complexity of the enumerative coding, based on this order, is
digits operations. A combination of the two
methods reduces the complexity on average by a constant factor.Comment: to appear in IEEE Transactions on Information Theor
Boolean complexes for Ferrers graphs
In this paper we provide an explicit formula for calculating the boolean
number of a Ferrers graph. By previous work of the last two authors, this
determines the homotopy type of the boolean complex of the graph. Specializing
to staircase shapes, we show that the boolean numbers of the associated Ferrers
graphs are the Genocchi numbers of the second kind, and obtain a relation
between the Legendre-Stirling numbers and the Genocchi numbers of the second
kind. In another application, we compute the boolean number of a complete
bipartite graph, corresponding to a rectangular Ferrers shape, which is
expressed in terms of the Stirling numbers of the second kind. Finally, we
analyze the complexity of calculating the boolean number of a Ferrers graph
using these results and show that it is a significant improvement over
calculating by edge recursion.Comment: final version, to appear in the The Australasian Journal of
Combinatoric
On k-crossings and k-nestings of permutations
We introduce k-crossings and k-nestings of permutations. We show that the
crossing number and the nesting number of permutations have a symmetric joint
distribution. As a corollary, the number of k-noncrossing permutations is equal
to the number of k-nonnesting permutations. We also provide some enumerative
results for k-noncrossing permutations for some values of k
Simplicial matrix-tree theorems
We generalize the definition and enumeration of spanning trees from the
setting of graphs to that of arbitrary-dimensional simplicial complexes
, extending an idea due to G. Kalai. We prove a simplicial version of
the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the
squares of the orders of their top-dimensional integral homology groups, in
terms of the Laplacian matrix of . As in the graphic case, one can
obtain a more finely weighted generating function for simplicial spanning trees
by assigning an indeterminate to each vertex of and replacing the
entries of the Laplacian with Laurent monomials. When is a shifted
complex, we give a combinatorial interpretation of the eigenvalues of its
weighted Laplacian and prove that they determine its set of faces uniquely,
generalizing known results about threshold graphs and unweighted Laplacian
eigenvalues of shifted complexes.Comment: 36 pages, 2 figures. Final version, to appear in Trans. Amer. Math.
So
Biconed graphs, edge-rooted forests, and h-vectors of matroid complexes
A well-known conjecture of Richard Stanley posits that the -vector of the
independence complex of a matroid is a pure -sequence. The
conjecture has been established for various classes but is open for graphic
matroids. A biconed graph is a graph with two specified `coning vertices', such
that every vertex of the graph is connected to at least one coning vertex. The
class of biconed graphs includes coned graphs, Ferrers graphs, and complete
multipartite graphs. We study the -vectors of graphic matroids arising from
biconed graphs, providing a combinatorial interpretation of their entries in
terms of `edge-rooted forests' of the underlying graph. This generalizes
constructions of Kook and Lee who studied the M\"obius coinvariant (the last
nonzero entry of the -vector) of graphic matroids of complete bipartite
graphs. We show that allowing for partially edge-rooted forests gives rise to a
pure multicomplex whose face count recovers the -vector, establishing
Stanley's conjecture for this class of matroids.Comment: 15 pages, 3 figures; V2: added omitted author to metadat
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