92 research outputs found

### On certain spaces of lattice diagram polynomials

The aim of this work is to study some lattice diagram determinants
$\Delta_L(X,Y)$. We recall that $M_L$ denotes the space of all partial
derivatives of $\Delta_L$. In this paper, we want to study the space
$M^k_{i,j}(X,Y)$ which is defined as the sum of $M_L$ spaces where the lattice
diagrams $L$ are obtained by removing $k$ cells from a given partition, these
cells being in the ``shadow'' of a given cell $(i,j)$ in a fixed Ferrers
diagram. We obtain an upper bound for the dimension of the resulting space
$M^k_{i,j}(X,Y)$, that we conjecture to be optimal. This dimension is a
multiple of $n!$ and thus we obtain a generalization of the $n!$ conjecture.
Moreover, these upper bounds associated to nice properties of some special
symmetric differential operators (the ``shift'' operators) allow us to
construct explicit bases in the case of one set of variables, i.e. for the
subspace $M^k_{i,j}(X)$ consisting of elements of 0 $Y$-degree

### Multivariate Fuss-Catalan numbers

Catalan numbers $C(n)=\frac{1}{n+1}{2n\choose n}$ enumerate binary trees and
Dyck paths. The distribution of paths with respect to their number $k$ of
factors is given by ballot numbers $B(n,k)=\frac{n-k}{n+k}{n+k\choose n}$.
These integers are known to satisfy simple recurrence, which may be visualised
in a ``Catalan triangle'', a lower-triangular two-dimensional array. It is
surprising that the extension of this construction to 3 dimensions generates
integers $B_3(n,k,l)$ that give a 2-parameter distribution of $C_3(n)=\frac 1
{2n+1} {3n\choose n}$, which may be called order-3 Fuss-Catalan numbers, and
enumerate ternary trees. The aim of this paper is a study of these integers
$B_3(n,k,l)$. We obtain an explicit formula and a description in terms of trees
and paths. Finally, we extend our construction to $p$-dimensional arrays, and
in this case we obtain a $(p-1)$-parameter distribution of $C_p(n)=\frac 1
{(p-1)n+1} {pn\choose n}$, the number of $p$-ary trees

### The # product in combinatorial Hopf algebras

We show that the # product of binary trees introduced by Aval and Viennot
[arXiv:0912.0798] is in fact defined at the level of the free associative
algebra, and can be extended to most of the classical combinatorial Hopf
algebras.Comment: 20 page

### Quasi-symmetric functions as polynomial functions on Young diagrams

We determine the most general form of a smooth function on Young diagrams,
that is, a polynomial in the interlacing or multirectangular coordinates whose
value depends only on the shape of the diagram. We prove that the algebra of
such functions is isomorphic to quasi-symmetric functions, and give a
noncommutative analog of this result.Comment: 34 pages, 4 figures, version including minor modifications suggested
by referee

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