1,709 research outputs found
Properties of four partial orders on standard Young tableaux
Let SYT_n be the set of all standard Young tableaux with n cells. After
recalling the definitions of four partial orders, the weak, KL, geometric and
chain orders on SYT_n and some of their crucial properties, we prove three main
results: (i)Intervals in any of these four orders essentially describe the
product in a Hopf algebra of tableaux defined by Poirier and Reutenauer. (ii)
The map sending a tableau to its descent set induces a homotopy equivalence of
the proper parts of all of these orders on tableaux with that of the Boolean
algebra 2^{[n-1]}. In particular, the M\"obius function of these orders on
tableaux is (-1)^{n-3}. (iii) For two of the four orders, one can define a more
general order on skew tableaux having fixed inner boundary, and similarly
analyze their homotopy type and M\"obius function.Comment: 24 pages, 3 figure
One-Dimensional Vertex Models Associated with a Class of Yangian Invariant Haldane-Shastry Like Spin Chains
We define a class of Yangian invariant Haldane-Shastry (HS)
like spin chains, by assuming that their partition functions can be written in
a particular form in terms of the super Schur polynomials. Using some
properties of the super Schur polynomials, we show that the partition functions
of this class of spin chains are equivalent to the partition functions of a
class of one-dimensional vertex models with appropriately defined energy
functions. We also establish a boson-fermion duality relation for the partition
functions of this class of supersymmetric HS like spin chains by using their
correspondence with one-dimensional vertex models
Cutoff for random to random card shuffle
In this paper, we use the eigenvalues of the random to random card shuffle to
prove a sharp upper bound for the total variation mixing time. Combined with
the lower bound due to Subag, we prove that this walk exhibits cutoff at
with window of order ,
answering a conjecture of Diaconis
A natural generalization of Balanced Tableaux
We introduce the notion of "type" of a tableau, that allows us to define new
families of tableaux including both balanced and standard Young tableaux. We
use these new objects to describe the set of reduced decompositions of any
permutation. We then generalize the work of Fomin \emph{et al.} by giving,
among other things, a new proof of the fact that balanced and standard tableaux
are equinumerous, and by exhibiting many new families of tableaux having
similar combinatorial properties to those of balanced tableaux.Comment: This new version cointains several major changes in order to take new
results into accoun
Stammering tableaux
The PASEP (Partially Asymmetric Simple Exclusion Process) is a probabilistic
model of moving particles, which is of great interest in combinatorics, since
it appeared that its partition function counts some tableaux. These tableaux
have several variants such as permutations tableaux, alternative tableaux,
tree- like tableaux, Dyck tableaux, etc. We introduce in this context certain
excursions in Young's lattice, that we call stammering tableaux (by analogy
with oscillating tableaux, vacillating tableaux, hesitating tableaux). Some
natural bijections make a link with rook placements in a double staircase,
chains of Dyck paths obtained by successive addition of ribbons, Laguerre
histories, Dyck tableaux, etc.Comment: Clarification and better exposition thanks reviewer's report
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