37,898 research outputs found
Permutation polynomials induced from permutations of subfields, and some complete sets of mutually orthogonal latin squares
We present a general technique for obtaining permutation polynomials over a
finite field from permutations of a subfield. By applying this technique to the
simplest classes of permutation polynomials on the subfield, we obtain several
new families of permutation polynomials. Some of these have the additional
property that both f(x) and f(x)+x induce permutations of the field, which has
combinatorial consequences. We use some of our permutation polynomials to
exhibit complete sets of mutually orthogonal latin squares. In addition, we
solve the open problem from a recent paper by Wu and Lin, and we give simpler
proofs of much more general versions of the results in two other recent papers.Comment: 13 pages; many new result
Orthogonality of Jack polynomials in superspace
Jack polynomials in superspace, orthogonal with respect to a
``combinatorial'' scalar product, are constructed. They are shown to coincide
with the Jack polynomials in superspace, orthogonal with respect to an
``analytical'' scalar product, introduced in hep-th/0209074 as eigenfunctions
of a supersymmetric quantum mechanical many-body problem. The results of this
article rely on generalizing (to include an extra parameter) the theory of
classical symmetric functions in superspace developed recently in
math.CO/0509408Comment: 22 pages, this supersedes the second part of math.CO/0412306; (v2) 24
pages, title and abstract slightly modified, minor changes, typos correcte
Correlation of arithmetic functions over
For a fixed polynomial , we study the number of polynomials of
degree over such that and are both
irreducible, an -analogue of the twin primes problem. In the
large- limit, we obtain a lower-order term for this count if we consider
non-monic polynomials, which depends on in a manner which is
consistent with the Hardy-Littlewood Conjecture. We obtain a saving of if
we consider monic polynomials only and is a scalar. To do this, we use
symmetries of the problem to get for free a small amount of averaging in
. This allows us to obtain additional saving from equidistribution
results for -functions. We do all this in a combinatorial framework that
applies to more general arithmetic functions than the indicator function of
irreducibles, including the M\"{o}bius function and divisor functions.Comment: Incorporated referee comments. Accepted for publication in Math.
Annale
String Reconstruction from Substring Compositions
Motivated by mass-spectrometry protein sequencing, we consider a
simply-stated problem of reconstructing a string from the multiset of its
substring compositions. We show that all strings of length 7, one less than a
prime, or one less than twice a prime, can be reconstructed uniquely up to
reversal. For all other lengths we show that reconstruction is not always
possible and provide sometimes-tight bounds on the largest number of strings
with given substring compositions. The lower bounds are derived by
combinatorial arguments and the upper bounds by algebraic considerations that
precisely characterize the set of strings with the same substring compositions
in terms of the factorization of bivariate polynomials. The problem can be
viewed as a combinatorial simplification of the turnpike problem, and its
solution may shed light on this long-standing problem as well. Using well known
results on transience of multi-dimensional random walks, we also provide a
reconstruction algorithm that reconstructs random strings over alphabets of
size in optimal near-quadratic time
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