Counting monomials

The weak Lefschetz property for monomial ideals of small type

In this work a combinatorial approach towards the weak Lefschetz property is developed that relates this property to enumerations of signed perfect matchings as well as to enumerations of signed families of non-intersecting lattice paths in certain triangular regions. This connection is used to study Artinian quotients by monomial ideals of a three-dimensional polynomial ring. Extending a main result in the recent memoir [Boij, Migliore, Miró-Roig, Nagel, Zanello], we completely classify the quotients of type two that have the weak Lefschetz property in characteristic zero. We also derive resu…

Counting Square free Cremona monomial maps

We use combinatorics tools to reobtain the classification of monomial quadratic Cremona transformations in any number of variables given in \cite{SV2} and to classify and count square free cubic Cremona maps with at most six variables, up to isomorphism.

Counting square free monomial cremona maps

We give a complete list of square-free Cremona maps with at most six variables, up to equivalence classes. We also build an algorithm to count monomial square-free Cremona transformations. Using this algorithm, we obtain a complete list of monomial square-free Cremona transformations in seven variables.

Enumerations of lozenge tilings, lattice paths, and perfect matchings and the weak Lefschetz property

MacMahon enumerated the plane partitions in an $a \times b \times c$ box. These are in bijection to lozenge tilings of a hexagon, to certain perfect matchings, and to families of non-intersecting lattice paths. In this work we consider more general regions, called triangular regions, and establish signed versions of the latter three bijections. Indeed, we use perfect matchings and families of non-intersecting lattice paths to define two signs of a lozenge tiling. A combinatorial argument involving a new method, called resolution of a puncture, then shows that the signs are in fact equivalent. …

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 chara…

计数半素数阶循环群上的Schur环

Counting Schur Rings over Cyclic Groups of Semi-prime Order

本文继续研究循环群上的Schur环计数问题，重点分析半素数阶循环群pq及4p阶循环群的结构与性质。