825 research outputs found
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
Separation probabilities for products of permutations
We study the mixing properties of permutations obtained as a product of two
uniformly random permutations of fixed cycle types. For instance, we give an
exact formula for the probability that elements are in distinct
cycles of the random permutation of obtained as product of two
uniformly random -cycles
Bijections and symmetries for the factorizations of the long cycle
We study the factorizations of the permutation into factors
of given cycle types. Using representation theory, Jackson obtained for each
an elegant formula for counting these factorizations according to the
number of cycles of each factor. In the cases Schaeffer and Vassilieva
gave a combinatorial proof of Jackson's formula, and Morales and Vassilieva
obtained more refined formulas exhibiting a surprising symmetry property. These
counting results are indicative of a rich combinatorial theory which has
remained elusive to this point, and it is the goal of this article to establish
a series of bijections which unveil some of the combinatorial properties of the
factorizations of into factors for all . We thereby obtain
refinements of Jackson's formulas which extend the cases treated by
Morales and Vassilieva. Our bijections are described in terms of
"constellations", which are graphs embedded in surfaces encoding the transitive
factorizations of permutations
Tame Decompositions and Collisions
A univariate polynomial f over a field is decomposable if f = g o h = g(h)
for nonlinear polynomials g and h. It is intuitively clear that the
decomposable polynomials form a small minority among all polynomials over a
finite field. The tame case, where the characteristic p of Fq does not divide n
= deg f, is fairly well-understood, and we have reasonable bounds on the number
of decomposables of degree n. Nevertheless, no exact formula is known if
has more than two prime factors. In order to count the decomposables, one wants
to know, under a suitable normalization, the number of collisions, where
essentially different (g, h) yield the same f. In the tame case, Ritt's Second
Theorem classifies all 2-collisions.
We introduce a normal form for multi-collisions of decompositions of
arbitrary length with exact description of the (non)uniqueness of the
parameters. We obtain an efficiently computable formula for the exact number of
such collisions at degree n over a finite field of characteristic coprime to p.
This leads to an algorithm for the exact number of decomposable polynomials at
degree n over a finite field Fq in the tame case
Matrix Factorizations and Representations of Quivers II: type ADE case
We study a triangulated category of graded matrix factorizations for a
polynomial of type ADE. We show that it is equivalent to the derived category
of finitely generated modules over the path algebra of the corresponding Dynkin
quiver. Also, we discuss a special stability condition for the triangulated
category in the sense of T. Bridgeland, which is naturally defined by the
grading.Comment: v2: typos corrected, added an appendix by K.Ued
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