On the Enumeration of Indexed Monomials and the Computation of Hilbert functions of Ladder Determinantal Varieties

# On the Enumeration of Indexed Monomials and the Computation of
Hilbert functions of Ladder Determinantal Varieties

**Sudhir R. Ghorpade**

**October 1, 1999**

### Abstract:

We outline the computation of an explicit formula for the Hilbert function of the ladder
determinantal varieties defined by the vanishing of all minors of a fixed
size of a rectangular matrix with indeterminate entries such that the
indeterminates in these minors are restricted to lie in some ladder shaped
region of the rectangular array.
Finding such a formula is equivalent to enumerating the set of monomials of a
fixed degree such that the support of these monomials is a subset of a `ladder' and
satisfies a certain ``index condition''.
We also describe applications of this formula for estimating the dimension
of ladder determinantal varieties.

Primary 05A15, 13C40, 13D40, 14M12; Secondary 05A19, 05E10, 14M15

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