29,169 research outputs found
A polyhedral model of partitions with bounded differences and a bijective proof of a theorem of Andrews, Beck, and Robbins
The main result of this paper is a bijective proof showing that the
generating function for partitions with bounded differences between largest and
smallest part is a rational function. This result is similar to the closely
related case of partitions with fixed differences between largest and smallest
parts which has recently been studied through analytic methods by Andrews,
Beck, and Robbins. Our approach is geometric: We model partitions with bounded
differences as lattice points in an infinite union of polyhedral cones.
Surprisingly, this infinite union tiles a single simplicial cone. This
construction then leads to a bijection that can be interpreted on a purely
combinatorial level.Comment: 12 pages, 5 figure
Mullineux involution and twisted affine Lie algebras
We use Naito-Sagaki's work [S. Naito & D. Sagaki, J. Algebra 245 (2001)
395--412, J. Algebra 251 (2002) 461--474] on Lakshmibai-Seshadri paths fixed by
diagram automorphisms to study the partitions fixed by Mullineux involution. We
characterize the set of Mullineux-fixed partitions in terms of crystal graphs
of basic representations of twisted affine Lie algebras of type
and of type . We set up bijections between
the set of symmetric partitions and the set of partitions into distinct parts.
We propose a notion of double restricted strict partitions. Bijections between
the set of restricted strict partitions (resp., the set of double restricted
strict partitions) and the set of Mullineux-fixed partitions in the odd case
(resp., in the even case) are obtained
Integrated information increases with fitness in the evolution of animats
One of the hallmarks of biological organisms is their ability to integrate
disparate information sources to optimize their behavior in complex
environments. How this capability can be quantified and related to the
functional complexity of an organism remains a challenging problem, in
particular since organismal functional complexity is not well-defined. We
present here several candidate measures that quantify information and
integration, and study their dependence on fitness as an artificial agent
("animat") evolves over thousands of generations to solve a navigation task in
a simple, simulated environment. We compare the ability of these measures to
predict high fitness with more conventional information-theoretic processing
measures. As the animat adapts by increasing its "fit" to the world,
information integration and processing increase commensurately along the
evolutionary line of descent. We suggest that the correlation of fitness with
information integration and with processing measures implies that high fitness
requires both information processing as well as integration, but that
information integration may be a better measure when the task requires memory.
A correlation of measures of information integration (but also information
processing) and fitness strongly suggests that these measures reflect the
functional complexity of the animat, and that such measures can be used to
quantify functional complexity even in the absence of fitness data.Comment: 27 pages, 8 figures, one supplementary figure. Three supplementary
video files available on request. Version commensurate with published text in
PLoS Comput. Bio
A bijection between Littlewood-Richardson tableaux and rigged configurations
A bijection is defined from Littlewood-Richardson tableaux to rigged
configurations. It is shown that this map preserves the appropriate statistics,
thereby proving a quasi-particle expression for the generalized Kostka
polynomials, which are q-analogues of multiplicities in tensor products of
irreducible general linear group modules indexed by rectangular partitions.Comment: 66 pages, AMS-LaTeX, requires xy.sty and related file
- β¦