23,320 research outputs found
Counting Combinatorial Choice Rules
I count the number of combinatorial choice rules that satisfy certain properties: Kelso-Crawford substitutability, and independence of irrelevant alternatives. The results are important for two-sided matching theory, where agents are modeled by combinatorial choice rules with these properties. The rules are a small, and asymptotically vanishing, fraction of all choice rules. But they are still exponentially more than the preference relations over individual agents- --which has positive implications for the Gale-Shapley algorithm of matching theory.Substitutability, Choice rules, Matching markets, Gale-Shapley Algorithm
Counting Combinatorial Choice Rules
I count the number of combinatorial choice rules that satisfy certain properties: Kelso-Crawford substitutability, and independence of irrelevant alternatives. The results are important for two-sided matching theory, where agents are modeled by combinatorial choice rules with these properties. The rules are a small, and asymtotically vanishing, fraction of all choice rules. But they are still exponentially more than the preference relations over individual agentsâwhich has positive implications for the Gale-Shapley algorithm of matching theory
Combinatorics in the Art of the Twentieth Century
This paper is motivated by a question I asked myself: How can combinatorial structures be used in a work of art? Immediately, other questions arose: Whether there are artists that work or think combinatorially? If so, what works have they produced in this way? What are the similarities and differences between art works produced using
combinatorics? This paper presents the first results of the attempt to answer these questions, being a survey of a selection of works that use or contain combinatorics in some way, including music, literature and visual arts, focusing on the twentieth century.Postprint (published version
Combinatorics of bicubic maps with hard particles
We present a purely combinatorial solution of the problem of enumerating
planar bicubic maps with hard particles. This is done by use of a bijection
with a particular class of blossom trees with particles, obtained by an
appropriate cutting of the maps. Although these trees have no simple local
characterization, we prove that their enumeration may be performed upon
introducing a larger class of "admissible" trees with possibly doubly-occupied
edges and summing them with appropriate signed weights. The proof relies on an
extension of the cutting procedure allowing for the presence on the maps of
special non-sectile edges. The admissible trees are characterized by simple
local rules, allowing eventually for an exact enumeration of planar bicubic
maps with hard particles. We also discuss generalizations for maps with
particles subject to more general exclusion rules and show how to re-derive the
enumeration of quartic maps with Ising spins in the present framework of
admissible trees. We finally comment on a possible interpretation in terms of
branching processes.Comment: 41 pages, 19 figures, tex, lanlmac, hyperbasics, epsf. Introduction
and discussion/conclusion extended, minor corrections, references adde
Combinatorial algebra for second-quantized Quantum Theory
We describe an algebra G of diagrams that faithfully gives a diagrammatic representation of the structures of both the HeisenbergâWeyl algebra H â the associative algebra of the creation and annihilation operators of quantum mechanics â and U(LH), the enveloping algebra of the Heisenberg Lie algebra LH. We show explicitly how G may be endowed with the structure of a Hopf algebra, which is also mirrored in the structure of U(LH). While both H and U(LH) are images of G, the algebra G has a richer structure and therefore embodies a finer combinatorial realization of the creationâannihilation system, of which it provides a concrete model
An update of quantum cohomology of homogeneous varieties
We describe recent progress on QH(G/P) with special emphasis of our own work.Comment: 18 pages. Any comments are welcom
Combinatorial Route to Algebra: The Art of Composition & Decomposition
We consider a general concept of composition and decomposition of objects,
and discuss a few natural properties one may expect from a reasonable choice
thereof. It will be demonstrated how this leads to multiplication and co-
multiplication laws, thereby providing a generic scheme furnishing
combinatorial classes with an algebraic structure. The paper is meant as a
gentle introduction to the concepts of composition and decomposition with the
emphasis on combinatorial origin of the ensuing algebraic constructions.Comment: 20 pages, 6 figure
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