• Speaker: Sergi Elizalde, Dartmouth College
• Title: Descents on quasi-Stirling permutations
• Date: 01/20/2021
• Time: 3:00 PM - 3:50 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Bruce E Sagan (bsagan@msu.edu)

Stirling permutations were introduced by Gessel and Stanley to give a combinatorial interpretation of certain polynomials related to Stirling numbers. A very natural extension of Stirling permutations are quasi-Stirling permutations, which are in bijection with labeled rooted plane trees. Archer et al. introduced these permutations, and conjectured that there are $(n+1)^{n-1}$ quasi-Stirling permutations of size $n$ having $n$ descents. In this talk we prove this conjecture. More generally, we give the generating function for quasi-Stirling permutations by the number of descents, which turns out to satisfy a beautiful equation involving Eulerian polynomials. We show that some of the properties of descents on usual permutations and on Stirling permutations have an analogue for quasi-Stirling permutations. Finally, we extend our results to a one-parameter family of permutations, called $k$-quasi-Stirling permutations, which are in bijection with certain decorated trees.

• Speaker: Richard Stanley, MIT
• Title: Two Analogues of Pascal's Triangle
• Date: 02/03/2021
• Time: 3:00 PM - 3:50 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Bruce E Sagan (bsagan@msu.edu)

Pascal's triangle is closely associated with the expansion of the product $(1+x)^n$. We will discuss two analogous arrays of numbers that are associated with the products $\prod_{i=0}^{n-1} \left(1+x^{2^i}+x^{2^{i+1}}\right)$ and $\prod_{i=1}^n \left(1+x^{F_{i+1}}\right)$, where $F_{i+1}$ is a Fibonacci number. All three arrays are special cases of a two-parameter family that might be interesting to investigate further.

• Speaker: Tom Roby, University of Connecticut
• Title: An action-packed introduction to homomesy
• Date: 02/17/2021
• Time: 3:00 PM - 3:50 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Bruce E Sagan (bsagan@msu.edu)

Dynamical Algebraic Combinatorics explores maps on sets of discrete combinatorial objects with particular attention to their orbit structure. Interesting counting questions immediately arise: How many orbits are there? What are their sizes? What is the period of the map if it's invertible? Are there any interesting statistics on the objects that are well-behaved under the map? One particular phenomenon of interest is ``homomesy'', where a statistic on the set of objects has the same average for each orbit of an action. Along with its intrinsic interest as a kind of hidden ``invariant'', homomesy can be used to help understand certain properties of the action. Proofs of homomesy often lead one to develop tools that further our understanding of the underlying dynamics, e.g., by finding an equivariant bijection. These notions can be lifted to higher (piecewise-linear and birational) realms, of which the combinatorial situation is a discrete shadow, and the resulting identities are somewhat surprising. Maps that can be decomposed as products of ``toggling'' involutions are particularly amenable to this line of analysis. This talk will be a introduction to these ideas, giving a number of examples.

• Speaker: Bridget Tenner, DePaul University
• Title: Odd diagram classes of permutations
• Date: 03/10/2021
• Time: 3:00 PM - 3:50 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Bruce E Sagan (bsagan@msu.edu)

Recently, Brenti and Carnevale introduced an odd analogue of the classical permutation diagram. This gives rise to odd analogues of other permutation aspects such as length and inversions. Unlike in the classical setting, multiple permutations might have the same odd diagram. This prompts the study of "odd diagram classes": sets of permutations having the same odd diagram. We will discuss the rich combinatorial structure of these classes, including connections to pattern avoiding permutations and the Bruhat order. This is joint work with Francesco Brenti and Angela Carnevale.

• Speaker: Jo Ellis-Monaghan, Korteweg-de Vries Instituut voor Wiskunde, Universiteit van Amsterdam
• Title: Graph theoretical tools for DNA self-assembly
• Date: 03/24/2021
• Time: 3:00 PM - 3:50 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Bruce E Sagan (bsagan@msu.edu)

Applications of immediate concern have driven some of the most interesting questions in the field of graph theory, for example graph drawing and computer chip layout problems, random graph theory and modeling the internet, graph connectivity measures and ecological systems, etc. Currently, scientists are engineering self-assembling DNA molecules to serve emergent applications in biomolecular computing, nanoelectronics, biosensors, drug delivery systems, and organic synthesis. Often, the self-assembled objects, e.g. lattices or polyhedral skeletons, may be modeled as graphs. Thus, these new technologies in self-assembly are now generating challenging new design problems for which graph theory is a natural tool. We will present some new applications in DNA self-assembly and describe some of the graph-theoretical design strategy problems arising from them. We will see how finding optimal design strategies leads to developing new algorithms for graphs, addressing new computational complexity questions, and finding new graph invariants corresponding to the minimum number of components necessary to build a target structure under various laboratory settings.

• Speaker: Volker Strehl, Friedrich-Alexander-Universität
• Title: A valuation problem for strict partitions — and where it comes from
• Date: 04/07/2021
• Time: 3:00 PM - 3:50 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Bruce E Sagan (bsagan@msu.edu)

I will present a rational function valuation on strict partitions as shapes that is defined via shifted standard tableaux. The goal is to determine explicit expressions for this valuation by solving a linear system that reflects the lattice structure of strict partitions. Somewhat surprisingly, this leads into the area of symmetric functions. The problem as such may appear as an isolated curiosity, yet is motivated by the extension of the model for an asymmetric annihilation process originally proposed by A. Ayyer and K. Mallick (2010).