Let P be a graded poset with 0 and 1 and rank at least 3. Assume that every rank 3 interval is a distributive lattice and that, for every interval of rank at least 4, the interval minus its endpoints is connected. It is shown that P is a distributive lattice, thus resolving an issue raised by Stanley. Similar theorems are proven for semimodular, modular, and complemented modular lattices. As a ...

For any partition λ let ω(λ) denote the four parameter weight ω(λ) = a P

This work-in-progress is intended as an exposition of the background material, results, and open problems of a particular poset theoretic study of Weyl characters and semisimple Lie algebra representations begun in the late 1970’s and early 1980’s in the work of Richard P. Stanley and Robert A. Proctor.

One of the most distinguishing characteristics Islam is fact that Holy Qur’an, has existed unaltered in its entirety for over fourteen hundred years. some people find this hard to believe and several have tried prove contrary. However, a close examination Qur’an process through which it been preserved clearly shows claim about Qur’an’s authenticity correct beyond doubt. The Quran refers oral re...

A distributive lattice L with 0 is finitary if every interval is finite. A function f : N0 N0 is a cover function for L if every element with n lower covers has f (n) upper covers. In this paper, all finitary distributive lattices with non-decreasing cover functions are characterized. A 1975 conjecture of Richard P. Stanley is thereby settled. 2000 Academic Press

Stanley Fish's recent book review' measures Richard Posner's The Problems of Jurisprudence2 more by the importance of the questions it raises than by the coherence of its answers. That standard enables Professor Fish to rank Judge Posner's book with the best of post-war legal thought (p 1475). The same standard demands a similar assessment of Fish's review. Few writings match the economy and pu...

We prove Stanley’s plethysm conjecture for the 2×n case, which composed with the work of Black and List provides another proof of Foulkes conjecture for the 2× n case. We also show that the way Stanley formulated his conjecture, it is false in general, and suggest an alternative formulation.

We present a general result that, using the theory of symmetric functions, produces several new classes of symmetric unimodal polynomials. The result has applications to enumerative combinatorics including the proof of a conjecture by R. Stanley.