957 research outputs found
Descent polynomials for permutations with bounded drop size
Motivated by juggling sequences and bubble sort, we examine permutations on
the set {1,2,...,n} with d descents and maximum drop size k. We give explicit
formulas for enumerating such permutations for given integers k and d. We also
derive the related generating functions and prove unimodality and symmetry of
the coefficients.Comment: 15 page
Carries, shuffling, and symmetric functions
The "carries" when n random numbers are added base b form a Markov chain with
an "amazing" transition matrix determined by Holte. This same Markov chain
occurs in following the number of descents or rising sequences when n cards are
repeatedly riffle shuffled. We give generating and symmetric function proofs
and determine the rate of convergence of this Markov chain to stationarity.
Similar results are given for type B shuffles. We also develop connections with
Gaussian autoregressive processes and the Veronese mapping of commutative
algebra.Comment: 23 page
Recurrences for Eulerian polynomials of type B and type D
We introduce new recurrences for the type B and type D Eulerian polynomials,
and interpret them combinatorially. These recurrences are analogous to a
well-known recurrence for the type A Eulerian polynomials. We also discuss
their relationship to polynomials introduced by Savage and Visontai in
connection to the real-rootedness of the corresponding Eulerian polynomials
An affine generalization of evacuation
We establish the existence of an involution on tabloids that is analogous to
Schutzenberger's evacuation map on standard Young tableaux. We find that the
number of its fixed points is given by evaluating a certain Green's polynomial
at , and satisfies a "domino-like" recurrence relation.Comment: 32 pages, 7 figure
Revstack sort, zigzag patterns, descent polynomials of -revstack sortable permutations, and Steingr\'imsson's sorting conjecture
In this paper we examine the sorting operator . Applying
this operator to a permutation is equivalent to passing the permutation
reversed through a stack. We prove theorems that characterise -revstack
sortability in terms of patterns in a permutation that we call
patterns. Using these theorems we characterise those permutations of length
which are sorted by applications of for . We
derive expressions for the descent polynomials of these six classes of
permutations and use this information to prove Steingr\'imsson's sorting
conjecture for those six values of . Symmetry and unimodality of the descent
polynomials for general -revstack sortable permutations is also proven and
three conjectures are given
- …