414 research outputs found
q-analog of tableau containment
We prove a -analog of the following result due to McKay, Morse and Wilf:
the probability that a random standard Young tableau of size contains a
fixed standard Young tableau of shape tends to
in the large limit, where is the number of
standard Young tableaux of shape . We also consider the probability
that a random pair of standard Young tableaux of the same shape
contains a fixed pair of standard Young tableaux.Comment: 20 pages, to appear J. Combin. Theory. Ser.
A Geometric Form for the Extended Patience Sorting Algorithm
Patience Sorting is a combinatorial algorithm that can be viewed as an
iterated, non-recursive form of the Schensted Insertion Algorithm. In recent
work the authors extended Patience Sorting to a full bijection between the
symmetric group and certain pairs of combinatorial objects (called pile
configurations) that are most naturally defined in terms of generalized
permutation pattern and barred pattern avoidance. This Extended Patience
Sorting Algorithm is very similar to the Robinson-Schensted-Knuth (or RSK)
Correspondence, which is itself built from repeated application of the
Schensted Insertion Algorithm.
In this work we introduce a geometric form for the Extended Patience Sorting
Algorithm that is in some sense a natural dual algorithm to G. Viennot's
celebrated Geometric RSK Algorithm. Unlike Geometric RSK, though, the lattice
paths coming from Patience Sorting are allowed to intersect. We thus also give
a characterization for the intersections of these lattice paths in terms of the
pile configurations associated with a given permutation under the Extended
Patience Sorting Algorithm.Comment: 14 pages, LaTeX, uses pstricks; v2: major revision after section 3;
to be published in Adv. Appl. Mat
Cyclic sieving and cluster multicomplexes
Reiner, Stanton, and White \cite{RSWCSP} proved results regarding the
enumeration of polygon dissections up to rotational symmetry. Eu and Fu
\cite{EuFu} generalized these results to Cartan-Killing types other than A by
means of actions of deformed Coxeter elements on cluster complexes of Fomin and
Zelevinsky \cite{FZY}. The Reiner-Stanton-White and Eu-Fu results were proven
using direct counting arguments. We give representation theoretic proofs of
closely related results using the notion of noncrossing and semi-noncrossing
tableaux due to Pylyavskyy \cite{PN} as well as some geometric realizations of
finite type cluster algebras due to Fomin and Zelevinsky \cite{FZClusterII}.Comment: To appear in Adv. Appl. Mat
q-Supernomial coefficients: From riggings to ribbons
q-Supernomial coefficients are generalizations of the q-binomial
coefficients. They can be defined as the coefficients of the Hall-Littlewood
symmetric function in a product of the complete symmetric functions or the
elementary symmetric functions. Hatayama et al. give explicit expressions for
these q-supernomial coefficients. A combinatorial expression as the generating
function of ribbon tableaux with (co)spin statistic follows from the work of
Lascoux, Leclerc and Thibon. In this paper we interpret the formulas by
Hatayama et al. in terms of rigged configurations and provide an explicit
statistic preserving bijection between rigged configurations and ribbon
tableaux thereby establishing a new direct link between these combinatorial
objects.Comment: 19 pages, svcon2e.sty file require
A sagbi basis for the quantum Grassmannian
The maximal minors of a p by (m + p) matrix of univariate polynomials of
degree n with indeterminate coefficients are themselves polynomials of degree
np. The subalgebra generated by their coefficients is the coordinate ring of
the quantum Grassmannian, a singular compactification of the space of rational
curves of degree np in the Grassmannian of p-planes in (m + p)-space. These
subalgebra generators are shown to form a sagbi basis. The resulting flat
deformation from the quantum Grassmannian to a toric variety gives a new
`Gr\"obner basis style' proof of the Ravi-Rosenthal-Wang formulas in quantum
Schubert calculus. The coordinate ring of the quantum Grassmannian is an
algebra with straightening law, which is normal, Cohen-Macaulay, Gorenstein and
Koszul, and the ideal of quantum Pl\"ucker relations has a quadratic Gr\"obner
basis. This holds more generally for skew quantum Schubert varieties. These
results are well-known for the classical Schubert varieties (n=0). We also show
that the row-consecutive p by p-minors of a generic matrix form a sagbi basis
and we give a quadratic Gr\"obner basis for their algebraic relations.Comment: 18 pages, 3 eps figure, uses epsf.sty. Dedicated to the memory of
Gian-Carlo Rot
- …