Journal of Statistical Planning and Inference
Volume 54, Issue 1 Abstract
2 September 1996
Pages 55-66


Young bitableaux, lattice paths and Hilbert functions

Sudhir R. Ghorpadea, *

a Department of Mathematics, Indian Institute of Technology, Powai Bombay 400076 India

Received 28 April 1994; revised 6 March 1995. Available online 20 February 1999.


A recent result on the enumeration of p-tuples of nonintersecting lattice paths in an integral rectangle is used to deduce a formula of Abhyankar for standard Young bitableaux of certain type, which gives the Hilbert function of a class of determinantal ideals. The lattice path formula is also shown to yield the numerator of the Hilbert series of these determinantal ideals and the h-vectors of the associated simplicial complexes. As a consequence, the a-invariant of these determinantal ideals is obtained in some cases, extending an earlier result of Gräbe. Some problems concerning generalizations of these results to `higher dimensions' are also discussed. In an appendix, the equivalence of Abhyankar's formula for unitableaux of a given shape and a formula of Hodge, obtained in connection with his determination of Hilbert functions of Schubert varieties in Grassmannians, is outlined.

Author Keywords: Standard tableaux; Nonintersecting lattice paths; Determinantal ideals; Hilbert functions; Stanley-Reisner ring; a-invariant

*Corresponding author.

Journal of Statistical Planning and Inference Abstract
Volume 54, Issue 1
2 September 1996
Pages 55-66

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