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On flushed partitions and concave compositions
In this work, we give combinatorial proofs for generating functions of two
problems, i.e., flushed partitions and concave compositions of even length. We
also give combinatorial interpretation of one problem posed by Sylvester
involving flushed partitions and then prove it. For these purposes, we first
describe an involution and use it to prove core identities. Using this
involution with modifications, we prove several problems of different nature,
including Andrews' partition identities involving initial repetitions and
partition theoretical interpretations of three mock theta functions of third
order , and . An identity of Ramanujan is proved
combinatorially. Several new identities are also established.Comment: 19 page
Spectral minimal partitions for a family of tori
We study partitions of the rectangular two-dimensional flat torus of length 1
and width b into k domains, with b a parameter in (0, 1] and k an integer. We
look for partitions which minimize the energy, definedas the largest first
eigenvalue of the Dirichlet Laplacian on the domains of the partition. We are
inparticular interested in the way these minimal partitions change when b is
varied. We present herean improvement, when k is odd, of the results on
transition values of b established by B. Helffer andT. Hoffmann-Ostenhof (2014)
and state a conjecture on those transition values. We establishan improved
upper bound of the minimal energy by explicitly constructing hexagonal tilings
of thetorus. These tilings are close to the partitions obtained from a
systematic numerical study based on an optimization algorithm adapted from B.
Bourdin, D. Bucur, and {\'E}. Oudet (2009). These numerical results also
support our conjecture concerning the transition values and give
betterestimates near those transition values
Notes on higher-dimensional partitions
We show the existence of a series of transforms that capture several
structures that underlie higher-dimensional partitions. These transforms lead
to a sequence of triangles whose entries are given combinatorial
interpretations as the number of particular types of skew Ferrers diagrams. The
end result of our analysis is the existence of a triangle, that we denote by F,
which implies that the data needed to compute the number of partitions of a
given positive integer is reduced by a factor of half. The number of spanning
rooted forests appears intriguingly in a family of entries in the triangle F.
Using modifications of an algorithm due to Bratley-McKay, we are able to
directly enumerate entries in some of the triangles. As a result, we have been
able to compute numbers of partitions of positive integers <= 25 in any
dimension.Comment: 36 pages; Mathematica file attached; See
http://www.physics.iitm.ac.in/~suresh/partitions.html to generate numbers of
partition
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