698 research outputs found

    Schur Partial Derivative Operators

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    A lattice diagram is a finite list L=((p_1,q_1),...,(p_n,q_n) of lattice cells. The corresponding lattice diagram determinant is \Delta_L(X;Y)=\det \| x_i^{p_j}y_i^{q_j} \|. These lattice diagram determinants are crucial in the study of the so-called ``n! conjecture'' of A. Garsia and M. Haiman. The space M_L is the space spanned by all partial derivatives of \Delta_L(X;Y). The ``shift operators'', which are particular partial symmetric derivative operators are very useful in the comprehension of the structure of the M_L spaces. We describe here how a Schur function partial derivative operator acts on lattice diagrams with distinct cells in the positive quadrant.Comment: 8 pages, LaTe

    On certain spaces of lattice diagram polynomials

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    The aim of this work is to study some lattice diagram determinants ΔL(X,Y)\Delta_L(X,Y). We recall that MLM_L denotes the space of all partial derivatives of ΔL\Delta_L. In this paper, we want to study the space Mi,jk(X,Y)M^k_{i,j}(X,Y) which is defined as the sum of MLM_L spaces where the lattice diagrams LL are obtained by removing kk cells from a given partition, these cells being in the ``shadow'' of a given cell (i,j)(i,j) in a fixed Ferrers diagram. We obtain an upper bound for the dimension of the resulting space Mi,jk(X,Y)M^k_{i,j}(X,Y), that we conjecture to be optimal. This dimension is a multiple of n!n! and thus we obtain a generalization of the n!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 Mi,jk(X)M^k_{i,j}(X) consisting of elements of 0 YY-degree

    Multivariate Fuss-Catalan numbers

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    Catalan numbers C(n)=1n+1(2nn)C(n)=\frac{1}{n+1}{2n\choose n} enumerate binary trees and Dyck paths. The distribution of paths with respect to their number kk of factors is given by ballot numbers B(n,k)=n−kn+k(n+kn)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 B3(n,k,l)B_3(n,k,l) that give a 2-parameter distribution of C3(n)=12n+1(3nn)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 B3(n,k,l)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 pp-dimensional arrays, and in this case we obtain a (p−1)(p-1)-parameter distribution of Cp(n)=1(p−1)n+1(pnn)C_p(n)=\frac 1 {(p-1)n+1} {pn\choose n}, the number of pp-ary trees
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