22,683 research outputs found
The complexity of counting poset and permutation patterns
We introduce a notion of pattern occurrence that generalizes both classical
permutation patterns as well as poset containment. Many questions about pattern
statistics and avoidance generalize naturally to this setting, and we focus on
functional complexity problems -- particularly those that arise by constraining
the order dimensions of the pattern and text posets. We show that counting the
number of induced, injective occurrences among dimension 2 posets is #P-hard;
enumerating the linear extensions that occur in realizers of dimension 2 posets
can be done in polynomial time, while for unconstrained dimension it is
GI-complete; counting not necessarily induced, injective occurrences among
dimension 2 posets is #P-hard; counting injective or not necessarily injective
occurrences of an arbitrary pattern in a dimension 1 text is #P-hard, although
it is in FP if the pattern poset is constrained to have bounded intrinsic
width; and counting injective occurrences of a dimension 1 pattern in an
arbitrary text is #P-hard, while it is in FP for bounded dimension texts. This
framework easily leads to a number of open questions, chief among which are (1)
is it #P-hard to count the number of occurrences of a dimension 2 pattern in a
dimension 1 text, and (2) is it #P-hard to count the number of texts which
avoid a given pattern?Comment: 15 page
Towards an optimised VLSI design algorithm for the constant matrix multiplication problem
The efficient design of multiplierless implementations of constant matrix multipliers is challenged by the huge solution search spaces even for small scale problems. Previous approaches tend to use hill-climbing algorithms risking sub-optimal results. The proposed algorithm avoids this by exploring parallel solutions. The computational complexity is tackled by modelling the problem in a format amenable to genetic programming and hardware acceleration. Results show an improvement on state of the art algorithms with future potential for even greater savings
Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams
We consider the problem of finding the number of matrices over a finite field
with a certain rank and with support that avoids a subset of the entries. These
matrices are a q-analogue of permutations with restricted positions (i.e., rook
placements). For general sets of entries these numbers of matrices are not
polynomials in q (Stembridge 98); however, when the set of entries is a Young
diagram, the numbers, up to a power of q-1, are polynomials with nonnegative
coefficients (Haglund 98).
In this paper, we give a number of conditions under which these numbers are
polynomials in q, or even polynomials with nonnegative integer coefficients. We
extend Haglund's result to complements of skew Young diagrams, and we apply
this result to the case when the set of entries is the Rothe diagram of a
permutation. In particular, we give a necessary and sufficient condition on the
permutation for its Rothe diagram to be the complement of a skew Young diagram
up to rearrangement of rows and columns. We end by giving conjectures
connecting invertible matrices whose support avoids a Rothe diagram and
Poincar\'e polynomials of the strong Bruhat order.Comment: 24 pages, 9 figures, 1 tabl
Staircases, dominoes, and the growth rate of 1324-avoiders
We establish a lower bound of 10.271 for the growth rate of the permutations avoiding 1324, and an upper bound of 13.5. This is done by first finding the precise growth rate of a subclass whose enumeration is related to West-2-stack-sortable permutations, and then combining copies of this subclass in particular ways
Distributions of several infinite families of mesh patterns
Br\"and\'en and Claesson introduced mesh patterns to provide explicit
expansions for certain permutation statistics as linear combinations of
(classical) permutation patterns. The first systematic study of avoidance of
mesh patterns was conducted by Hilmarsson et al., while the first systematic
study of the distribution of mesh patterns was conducted by the first two
authors.
In this paper, we provide far-reaching generalizations for 8 known
distribution results and 5 known avoidance results related to mesh patterns by
giving distribution or avoidance formulas for certain infinite families of mesh
patterns in terms of distribution or avoidance formulas for smaller patterns.
Moreover, as a corollary to a general result, we find the distribution of one
more mesh pattern of length 2.Comment: 27 page
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