38,379 research outputs found
On the Intriguing Problem of Counting (n+1,n+2)-Core Partitions into Odd Parts
Tewodros Amdeberhan and Armin Straub initiated the study of enumerating
subfamilies of the set of (s,t)-core partitions. While the enumeration of
(n+1,n+2)-core partitions into distinct parts is relatively easy (in fact it
equals the Fibonacci number F_{n+2}), the enumeration of (n+1,n+2)-core
partitions into odd parts remains elusive.
Straub computed the first eleven terms of that sequence, and asked for a
"formula," or at least a fast way, to compute many terms. While we are unable
to find a "fast" algorithm, we did manage to find a "faster" algorithm, which
enabled us to compute 23 terms of this intriguing sequence. We strongly believe
that this sequence has an algebraic generating function, since a "sister
sequence" (see the article), is OEIS sequence A047749 that does have an
algebraic generating function. One of us (DZ) is pledging a donation of 100
dollars to the OEIS, in honor of the first person to generate sufficiently many
terms to conjecture (and prove non-rigorously) an algebraic equation for the
generating function of this sequence, and another 100 dollars for a rigorous
proof of that conjecture.
Finally, we also develop algorithms that find explicit generating functions
for other, more tractable, families of (n+1,n+2)-core partitions.Comment: 12 pages, accompanied by Maple package. This version announces that
our questions were all answered by Paul Johnson, and a donation to the OEIS,
in his honor, has been mad
Computing Exact Clustering Posteriors with Subset Convolution
An exponential-time exact algorithm is provided for the task of clustering n
items of data into k clusters. Instead of seeking one partition, posterior
probabilities are computed for summary statistics: the number of clusters, and
pairwise co-occurrence. The method is based on subset convolution, and yields
the posterior distribution for the number of clusters in O(n * 3^n) operations,
or O(n^3 * 2^n) using fast subset convolution. Pairwise co-occurrence
probabilities are then obtained in O(n^3 * 2^n) operations. This is
considerably faster than exhaustive enumeration of all partitions.Comment: 6 figure
A Fast Algorithm for MacMahon's Partition Analysis
This paper deals with evaluating constant terms of a special class of
rational functions, the Elliott-rational functions. The constant term of such a
function can be read off immediately from its partial fraction decomposition.
We combine the theory of iterated Laurent series and a new algorithm for
partial fraction decompositions to obtain a fast algorithm for MacMahon's Omega
calculus, which (partially) avoids the "run-time explosion" problem when
eliminating several variables. We discuss the efficiency of our algorithm by
investigating problems studied by Andrews and his coauthors; our running time
is much less than that of their Omega package.Comment: 22 page
Approximately Sampling Elements with Fixed Rank in Graded Posets
Graded posets frequently arise throughout combinatorics, where it is natural
to try to count the number of elements of a fixed rank. These counting problems
are often -complete, so we consider approximation algorithms for
counting and uniform sampling. We show that for certain classes of posets,
biased Markov chains that walk along edges of their Hasse diagrams allow us to
approximately generate samples with any fixed rank in expected polynomial time.
Our arguments do not rely on the typical proofs of log-concavity, which are
used to construct a stationary distribution with a specific mode in order to
give a lower bound on the probability of outputting an element of the desired
rank. Instead, we infer this directly from bounds on the mixing time of the
chains through a method we call .
A noteworthy application of our method is sampling restricted classes of
integer partitions of . We give the first provably efficient Markov chain
algorithm to uniformly sample integer partitions of from general restricted
classes. Several observations allow us to improve the efficiency of this chain
to require space, and for unrestricted integer partitions,
expected time. Related applications include sampling permutations
with a fixed number of inversions and lozenge tilings on the triangular lattice
with a fixed average height.Comment: 23 pages, 12 figure
Random sampling of plane partitions
This article presents uniform random generators of plane partitions according
to the size (the number of cubes in the 3D interpretation). Combining a
bijection of Pak with the method of Boltzmann sampling, we obtain random
samplers that are slightly superlinear: the complexity is in
approximate-size sampling and in exact-size sampling
(under a real-arithmetic computation model). To our knowledge, these are the
first polynomial-time samplers for plane partitions according to the size
(there exist polynomial-time samplers of another type, which draw plane
partitions that fit inside a fixed bounding box). The same principles yield
efficient samplers for -boxed plane partitions (plane partitions
with two dimensions bounded), and for skew plane partitions. The random
samplers allow us to perform simulations and observe limit shapes and frozen
boundaries, which have been analysed recently by Cerf and Kenyon for plane
partitions, and by Okounkov and Reshetikhin for skew plane partitions.Comment: 23 page
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