62,759 research outputs found
A Spectral Approach to Consecutive Pattern-Avoiding Permutations
We consider the problem of enumerating permutations in the symmetric group on
elements which avoid a given set of consecutive pattern , and in
particular computing asymptotics as tends to infinity. We develop a general
method which solves this enumeration problem using the spectral theory of
integral operators on , where the patterns in has length
. Kre\u{\i}n and Rutman's generalization of the Perron--Frobenius theory
of non-negative matrices plays a central role. Our methods give detailed
asymptotic expansions and allow for explicit computation of leading terms in
many cases. As a corollary to our results, we settle a conjecture of Warlimont
on asymptotics for the number of permutations avoiding a consecutive pattern.Comment: a reference is added; corrected typos; to appear in Journal of
Combinatoric
Motzkin Intervals and Valid Hook Configurations
We define a new natural partial order on Motzkin paths that serves as an
intermediate step between two previously-studied partial orders. We provide a
bijection between valid hook configurations of -avoiding permutations and
intervals in these new posets. We also show that valid hook configurations of
permutations avoiding (or equivalently, ) are counted by the same
numbers that count intervals in the Motzkin-Tamari posets that Fang recently
introduced, and we give an asymptotic formula for these numbers. We then
proceed to enumerate valid hook configurations of permutations avoiding other
collections of patterns. We also provide enumerative conjectures, one of which
links valid hook configurations of -avoiding permutations, intervals in
the new posets we have defined, and certain closed lattice walks with small
steps that are confined to a quarter plane.Comment: 22 pages, 8 figure
The number of inversions of permutations with fixed shape
The Robinson-Schensted correspondence can be viewed as a map from
permutations to partitions. In this work, we study the number of inversions of
permutations corresponding to a fixed partition under this map.
Hohlweg characterized permutations having shape with the minimum
number of inversions. Here, we give the first results in this direction for
higher numbers of inversions. We give explicit conjectures for both the
structure and the number of permutations associated to where the
extra number of inversions is less than the length of the smallest column of
. We prove the result when has two columns.Comment: 19 pages, 2 figure
Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations
Defant, Engen, and Miller defined a permutation to be uniquely sorted if it
has exactly one preimage under West's stack-sorting map. We enumerate classes
of uniquely sorted permutations that avoid a pattern of length three and a
pattern of length four by establishing bijections between these classes and
various lattice paths. This allows us to prove nine conjectures of Defant.Comment: 18 pages, 16 figures, new version with updated abstract and
reference
On the M\"obius Function of Permutations With One Descent
The set of all permutations, ordered by pattern containment, is a poset. We
give a formula for the M\"obius function of intervals in this poset,
for any permutation with at most one descent. We compute the M\"obius
function as a function of the number and positions of pairs of consecutive
letters in that are consecutive in value. As a result of this we show
that the M\"obius function is unbounded on the poset of all permutations. We
show that the M\"obius function is zero on any interval where
has a triple of consecutive letters whose values are consecutive and monotone.
We also conjecture values of the M\"obius function on some other intervals of
permutations with at most one descent
Bijective enumeration of some colored permutations given by the product of two long cycles
Let be the permutation on symbols defined by $\gamma_n = (1\
2\...\ n)\betannp\gamma_n \beta^{-1}\frac{1}{n- p+1}\alpha\gamma_n\alphamn+1$, an
unexpected connection previously found by several authors by means of algebraic
methods. Moreover, our bijection allows us to refine the latter result with the
cycle type of the permutations.Comment: 22 pages. Version 1 is a short version of 12 pages, entitled "Linear
coefficients of Kerov's polynomials: bijective proof and refinement of
Zagier's result", published in DMTCS proceedings of FPSAC 2010, AN, 713-72
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