DOUBLE FEATURE!!

TWO TALKS FROM: The Many Combinatorial Legacies of Richard P. Stanley: Immense Birthday Glory of the Epic Catalonian Rascal

Speaker:   Jim Propp, U Mass Lowell

Title :        The further adventures of Stanley's transfer map

Date:         Monday, March 3, 2025 at 2 pm in 745 POT

Abstract:

In 1986 Stanley showed that the natural bijection between antichains and order ideals of a poset P gives rise
to a well-behaved continuous piecewise-linear map from the order polytope of P to a new polytope he called
the chain polytope of P.  I’ll describe several directions in which this map has been generalized and also discuss
the role these newer maps have played in dynamical algebraic combinatorics.

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Speaker:   Sara Billey, University of Washington

Title :        Brewing Fubini-Bruhat partial orders with ASMs

Date:         Monday, March 3, 2025 at 2:20 pm in 745 POT

Abstract:

Fubini words (also known as Cayley permutations, packed words, and surjective words) are generalized permutations, allowing for repeated letters, and they are in one-to-one correspondence with ordered set partitions.  Brendan Pawlowski and Brendon Rhoades extended permutation matrices to pattern matrices for Fubini words.  Under a lower triangular action, these pattern matrices produce cells in projective space, specifically (Pk−1)n.  The containment of the cell closures in the Zariski topology gives rise to a poset which generalizes the Bruhat order for permutations. Unlike Bruhat order, containment is not equivalent to intersection of a cell with the closure of another cell. This allows for a refinement of the poset. 

It is additionally possible to define a weaker order, giving rise to a subposet containing all the elements. We call these orders, in order of decreasing strength, the espresso, medium roast, and decaf Fubini-Bruhat orders. The espresso and medium roast orders are not ranked in general. The decaf order is ranked by codimension of the corresponding cells. In fact, the decaf order has rank generating function given by a well-known q-analog of the Stirling numbers of the second kind.

In this paper, we give increasingly smaller sets of equations describing the cell closures, which lead to several different combinatorial descriptions for the relations in all three orders. Studying these partial orders and their associated rank functions has lead to generalizations of Fulton’s essential set and alternating sign matrices.

This talk is based on joint work with Stark Ryan and Matjaˇz Konvalinka.