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).

[1]  Holger Brenner,et al.  A note on the weak Lefschetz property of monomial complete intersections in positive characteristic , 2010, 1003.0824.

[2]  Junzo Watanabe,et al.  The Dilworth Number of Artinian Rings and Finite Posets with Rank Function , 1987 .

[3]  U. Nagel,et al.  Monomial ideals, almost complete intersections and the Weak Lefschetz property , 2008, 0811.1023.

[4]  M. Bona.  Introduction to Enumerative Combinatorics , 2005 .

[5]  Charalambos A. Charalambides,et al.  Enumerative combinatorics , 2018, SIGA.

[6]  Christian Krattenthaler.  Another Involution Principle-Free Bijective Proof of Stanley's Hook-Content Formula , 1999, J. Comb. Theory, Ser. A.

[7]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[8]  B. Sturmfels,et al.  Combinatorial Commutative Algebra , 2004 .

[9]  Les Reid,et al.  On Complete Intersections and Their Hilbert Functions , 1991, Canadian Mathematical Bulletin.

[10]  T. Harima,et al.  The Weak and Strong Lefschetz properties for Artinian K-algebras , 2002, math/0208201.

[11]  C. Krattenthaler.  ADVANCED DETERMINANT CALCULUS , 1999, math/9902004.

[12]  R. Stanley.  Hilbert functions of graded algebras , 1978 .

[13]  Masao Hara,et al.  The determinants of certain matrices arising from the Boolean lattice , 2008, Discret. Math..

[14]  Paul Zinn-Justin,et al.  A Bijection Between Classes of Fully Packed Loops and Plane Partitions , 2004, Electron. J. Comb..

[15]  Emil Grosswald,et al.  The Theory of Partitions , 1984 .

[16]  F. Zanello,et al.  Forcing the strong Lefschetz and the maximal rank properties , 2008, 0804.2877.

[17]  H. Halberstam,et al.  Combinatorial Analysis , 1971, Nature.

[18]  The Hilbert functions which force the weak Lefschetz property , 2006, math/0609150.

[19]  R. Stanley.  Enumerative Combinatorics: Volume 1 , 2011 .

[20]  Donald E. Knuth,et al.  Overlapping Pfaffians , 1995, Electron. J. Comb..

[21]  D. Cook,et al.  The weak Lefschetz property, monomial ideals, and lozenges , 2009, 0909.3509.

[22]  Igor Pak,et al.  Partition bijections, a survey , 2006 .

[23]  W. J. Thron,et al.  Encyclopedia of Mathematics and its Applications. , 1982 .

[24]  Frank Ruskey,et al.  Transition Restricted Gray Codes , 1996, Electron. J. Comb..

[25]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[26]  G. Andrews,et al.  Integer Partitions: Ferrers graphs , 2004 .

[27]  Carlos Tomei,et al.  The Problem of the Calissons , 1989 .