Fabrizio Zanello

Monomial complete intersections, the weak Lefschetz property and plane partitions

We characterize the monomial complete intersections in three variables satisfying the Weak Lefschetz Property (WLP), as a function of the characteristic of the base field. Our result presents a surprising, and still combinatorially obscure, connection with the enumeration of plane partitions. It turns out that the rational primes p dividing the number, M(a,b,c), of plane partitions contained inside an arbitrary box of given sides a,b,c are precisely those for which a suitable monomial complete intersection (explicitly constructed as a bijective function of a,b,c) fails to have the WLP in characteristic p. We wonder how powerful can be this connection between combinatorial commutative algebra and partition theory. We present a first result in this direction, by deducing, using our algebraic techniques for the WLP, some explicit information on the rational primes dividing M(a,b,c).

The Catalan Case of Armstrong's Conjecture on Simultaneous Core Partitions

A beautiful recent conjecture of D. Armstrong predicts the average size of a partition that is simultaneously an

On the degree two entry of a Gorenstein

s

h

-core and a

-vector and a conjecture of Stanley

t

Level algebras with bad properties

-core, where

Stanley’s theorem on codimension 3 Gorenstein ℎ-vectors

s

Unimodality of partitions with distinct parts inside Ferrers shapes

and

Extending the idea of compressed algebra to arbitrary socle-vectors

t

Zeilberger’s KOH theorem and the strict unimodality of q-binomial coefficients

are coprime. Our goal is to prove this conjecture when