909 research outputs found

### Lattice Diagram Polynomials and Extended Pieri Rules

The lattice cell in the ${i+1}^{st}$ row and ${j+1}^{st}$ column of the
positive quadrant of the plane is denoted $(i,j)$. If $\mu$ is a partition of
$n+1$, we denote by $\mu/ij$ the diagram obtained by removing the cell $(i,j)$
from the (French) Ferrers diagram of $\mu$. We set $\Delta_{\mu/ij}=\det \|
x_i^{p_j}y_i^{q_j} \|_{i,j=1}^n$, where $(p_1,q_1),... ,(p_n,q_n)$ are the
cells of $\mu/ij$, and let ${\bf M}_{\mu/ij}$ be the linear span of the partial
derivatives of $\Delta_{\mu/ij}$. The bihomogeneity of $\Delta_{\mu/ij}$ and
its alternating nature under the diagonal action of $S_n$ gives ${\bf
M}_{\mu/ij}$ the structure of a bigraded $S_n$-module. We conjecture that ${\bf
M}_{\mu/ij}$ is always a direct sum of $k$ left regular representations of
$S_n$, where $k$ is the number of cells that are weakly north and east of
$(i,j)$ in $\mu$. We also make a number of conjectures describing the precise
nature of the bivariate Frobenius characteristic of ${\bf M}_{\mu/ij}$ in terms
of the theory of Macdonald polynomials. On the validity of these conjectures,
we derive a number of surprising identities. In particular, we obtain a
representation theoretical interpretation of the coefficients appearing in some
Macdonald Pieri Rules.Comment: 77 pages, Te

### Orthonormal Systems in Linear Spans

We show that any $N$-dimensional linear subspace of $L^2(\mathbb{T})$ admits
an orthonormal system such that the $L^2$ norm of the square variation operator
$V^2$ is as small as possible. When applied to the span of the trigonometric
system, we obtain an orthonormal system of trigonometric polynomials with a
$V^2$ operator that is considerably smaller than the associated operator for
the trigonometric system itself.Comment: 18 page

### Parallelogram polyominoes, the sandpile model on a complete bipartite graph, and a q,t-Narayana polynomial

We classify recurrent configurations of the sandpile model on the complete
bipartite graph K_{m,n} in which one designated vertex is a sink. We present a
bijection from these recurrent configurations to decorated parallelogram
polyominoes whose bounding box is a m*n rectangle. Several special types of
recurrent configurations and their properties via this bijection are examined.
For example, recurrent configurations whose sum of heights is minimal are shown
to correspond to polyominoes of least area. Two other classes of recurrent
configurations are shown to be related to bicomposition matrices, a matrix
analogue of set partitions, and (2+2)-free partially ordered sets.
A canonical toppling process for recurrent configurations gives rise to a
path within the associated parallelogram polyominoes. This path bounces off the
external edges of the polyomino, and is reminiscent of Haglund's well-known
bounce statistic for Dyck paths. We define a collection of polynomials that we
call q,t-Narayana polynomials, defined to be the generating function of the
bistatistic (area,parabounce) on the set of parallelogram polyominoes, akin to
the (area,hagbounce) bistatistic defined on Dyck paths in Haglund (2003). In
doing so, we have extended a bistatistic of Egge, Haglund, Kremer and
Killpatrick (2003) to the set of parallelogram polyominoes. This is one answer
to their question concerning extensions to other combinatorial objects.
We conjecture the q,t-Narayana polynomials to be symmetric and prove this
conjecture for numerous special cases. We also show a relationship between
Haglund's (area,hagbounce) statistic on Dyck paths, and our bistatistic
(area,parabounce) on a sub-collection of those parallelogram polyominoes living
in a (n+1)*n rectangle

### Combinatorics of Labelled Parallelogram polyominoes

We obtain explicit formulas for the enumeration of labelled parallelogram
polyominoes. These are the polyominoes that are bounded, above and below, by
north-east lattice paths going from the origin to a point (k,n). The numbers
from 1 and n (the labels) are bijectively attached to the $n$ north steps of
the above-bounding path, with the condition that they appear in increasing
values along consecutive north steps. We calculate the Frobenius characteristic
of the action of the symmetric group S_n on these labels. All these enumeration
results are refined to take into account the area of these polyominoes. We make
a connection between our enumeration results and the theory of operators for
which the intergral Macdonald polynomials are joint eigenfunctions. We also
explain how these same polyominoes can be used to explicitly construct a linear
basis of a ring of SL_2-invariants.Comment: 25 pages, 9 figure

### Lattice Diagram polynomials in one set of variables

The space $M_{\mu/i,j}$ spanned by all partial derivatives of the lattice
polynomial $\Delta_{\mu/i,j}(X;Y)$ is investigated in math.CO/9809126 and many
conjectures are given. Here, we prove all these conjectures for the $Y$-free
component $M_{\mu/i,j}^0$ of $M_{\mu/i,j}$. In particular, we give an explicit
bases for $M_{\mu/i,j}^0$ which allow us to prove directly the central {\sl
four term recurrence} for these spaces.Comment: 15 page

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