856 research outputs found
Algebraic aspects of increasing subsequences
We present a number of results relating partial Cauchy-Littlewood sums,
integrals over the compact classical groups, and increasing subsequences of
permutations. These include: integral formulae for the distribution of the
longest increasing subsequence of a random involution with constrained number
of fixed points; new formulae for partial Cauchy-Littlewood sums, as well as
new proofs of old formulae; relations of these expressions to orthogonal
polynomials on the unit circle; and explicit bases for invariant spaces of the
classical groups, together with appropriate generalizations of the
straightening algorithm.Comment: LaTeX+amsmath+eepic; 52 pages. Expanded introduction, new references,
other minor change
On the Distributions of the Lengths of the Longest Monotone Subsequences in Random Words
We consider the distributions of the lengths of the longest weakly increasing
and strongly decreasing subsequences in words of length N from an alphabet of k
letters. We find Toeplitz determinant representations for the exponential
generating functions (on N) of these distribution functions and show that they
are expressible in terms of solutions of Painlev\'e V equations. We show
further that in the weakly increasing case the generating function gives the
distribution of the smallest eigenvalue in the k x k Laguerre random matrix
ensemble and that the distribution itself has, after centering and normalizing,
an N -> infinity limit which is equal to the distribution function for the
largest eigenvalue in the Gaussian Unitary Ensemble of k x k hermitian matrices
of trace zero.Comment: 30 pages, revised version corrects an error in the statement of
Theorem
Statistical Self-Similarity of One-Dimensional Growth Processes
For one-dimensional growth processes we consider the distribution of the
height above a given point of the substrate and study its scale invariance in
the limit of large times. We argue that for self-similar growth from a single
seed the universal distribution is the Tracy-Widom distribution from the theory
of random matrices and that for growth from a flat substrate it is some other,
only numerically determined distribution. In particular, for the polynuclear
growth model in the droplet geometry the height maps onto the longest
increasing subsequence of a random permutation, from which the height
distribution is identified as the Tracy-Widom distribution.Comment: 11 pages, iopart, epsf, 2 postscript figures, submitted to Physica A,
in an Addendum the distribution for the flat case is identified analyticall
Combinatorics of patience sorting piles
Despite having been introduced in 1962 by C.L. Mallows, the combinatorial
algorithm Patience Sorting is only now beginning to receive significant
attention due to such recent deep results as the Baik-Deift-Johansson Theorem
that connect it to fields including Probabilistic Combinatorics and Random
Matrix Theory.
The aim of this work is to develop some of the more basic combinatorics of
the Patience Sorting Algorithm. In particular, we exploit the similarities
between Patience Sorting and the Schensted Insertion Algorithm in order to do
things that include defining an analog of the Knuth relations and extending
Patience Sorting to a bijection between permutations and certain pairs of set
partitions. As an application of these constructions we characterize and
enumerate the set S_n(3-\bar{1}-42) of permutations that avoid the generalized
permutation pattern 2-31 unless it is part of the generalized pattern 3-1-42.Comment: 19 pages, LaTeX; uses pstricks; view PS, not DVI; use dvips + ps2pdf,
not dvi2pdf; part of FPSAC'05 proceedings; v3: final journal version, revised
Section 3.
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