1 5 D ec 1 99 9 FINITE POSETS AND FERRERS SHAPES

Finite Posets and Ferrers Shapes

Unimodality of partitions with distinct parts inside Ferrers shapes

We investigate the rank-generating function Fλ of the poset of partitions contained inside a given shifted Ferrers shape λ. When λ has four parts, we show that Fλ is unimodal when λ = 〈n, n − 1, n − 2, n − 3〉, for any n ≥ 4, and that unimodality fails for the doubly-indexed, infinite family of partitions of the form λ = 〈n, n− t, n− 2t, n− 3t〉, for any given t ≥ 2 and n large enough with respec...

A Bijection between Maximal Chains in Fibonacci Posets

In a 1988 paper [9], Stanley introduced a class of partially ordered sets, called differential posets, defined independently by S. Fomin [1] who called them Y-graphs. The prototypical example of a differential poset is Young's lattice, Y, the lattice of integer partitions ordered by inclusion of Ferrers diagrams. Another important example is given by the Fibonacci r-differential poset, Z(r), de...

Rook Poset Equivalence of Ferrers Boards

A natural construction due to K. Ding yields Schubert varieties from Ferrers boards. The poset structure of the Schubert cells in these varieties is equal to the poset of maximal rook placements on the Ferrers board under the Bruhat order. We determine when two Ferrers boards have isomorphic rook posets. Equivalently, we give an exact categorization of when two Ding Schubert varieties have iden...

On Operators on Polynomials Preserving Real-Rootedness and the Neggers-Stanley Conjecture

We refine a technique used in a paper by Schur on real-rooted polynomials. This amounts to an extension of a theorem of Wagner on Hadamard products of Pólya frequency sequences. We also apply our results to polynomials for which the Neggers-Stanley Conjecture is known to hold. More precisely, we settle interlacing properties for E-polynomials of series-parallel posets and column-strict labelled...