6,395 research outputs found
Asymptotic distribution of fixed points of pattern-avoiding involutions
For a variety of pattern-avoiding classes, we describe the limiting
distribution for the number of fixed points for involutions chosen uniformly at
random from that class. In particular we consider monotone patterns of
arbitrary length as well as all patterns of length 3. For monotone patterns we
utilize the connection with standard Young tableaux with at most rows and
involutions avoiding a monotone pattern of length . For every pattern of
length 3 we give the bivariate generating function with respect to fixed points
for the involutions that avoid that pattern, and where applicable apply tools
from analytic combinatorics to extract information about the limiting
distribution from the generating function. Many well-known distributions
appear.Comment: 16 page
A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics
A ``hybrid method'', dedicated to asymptotic coefficient extraction in
combinatorial generating functions, is presented, which combines Darboux's
method and singularity analysis theory. This hybrid method applies to functions
that remain of moderate growth near the unit circle and satisfy suitable
smoothness assumptions--this, even in the case when the unit circle is a
natural boundary. A prime application is to coefficients of several types of
infinite product generating functions, for which full asymptotic expansions
(involving periodic fluctuations at higher orders) can be derived. Examples
relative to permutations, trees, and polynomials over finite fields are treated
in this way.Comment: 31 page
On the Spectrum of the Derangement Graph
We derive several interesting formulae for the eigenvalues of the derangement graph and use them to settle affirmatively a conjecture of Ku regarding the least eigenvalue
Real Zeros and Partitions without singleton blocks
We prove that the generating polynomials of partitions of an -element set
into non-singleton blocks, counted by the number of blocks, have real roots
only and we study the asymptotic behavior of the leftmost roots. We apply this
information to find the most likely number of blocks.Comment: 16 page
Binomial Eulerian polynomials for colored permutations
Binomial Eulerian polynomials first appeared in work of Postnikov, Reiner and
Williams on the face enumeration of generalized permutohedra. They are
-positive (in particular, palindromic and unimodal) polynomials which
can be interpreted as -polynomials of certain flag simplicial polytopes and
which admit interesting Schur -positive symmetric function
generalizations. This paper introduces analogues of these polynomials for
-colored permutations with similar properties and uncovers some new
instances of equivariant -positivity in geometric combinatorics.Comment: Final version; minor change
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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