Combinatorial Potlatch 2016
Seattle University
Saturday, November 19, 2016

The Combinatorial Potlatch is an irregularly scheduled, floating, one-day conference. It has been held for many years at various locations around Puget Sound and southern British Columbia, and is an opportunity for combinatorialists in the region to gather informally for a day of invited talks and conversation. While most who attend work in, or near, the Puget Sound basin, all are welcome.  Typically there are three talks given by speakers who are visiting or new to the area, along with breaks for coffee and lunch. Many participants remain for dinner at a local restaurant or pub.

The American Heritage Dictionary defines "potlatch" as: A ceremonial feast among certain Native American peoples of the northwest Pacific coast, as in celebration of a marriage or an accession, at which the host distributes gifts according to each guest's rank or status. Between rival groups the potlatch could involve extravagant or competitive giving and destruction by the host of valued items as a display of superior wealth. [Chinook Jargon, from Nootka p'achitl, to make a potlatch gift.]

This fall's Potlatch is being hosted by the Department of Mathematics at Seattle University in central Seattle, Washington on Saturday, November 19, 2016.

### Schedule

All talks will be held in Casey 500, with registration and breaks nearby. See the Getting There section for exact locations and directions.

The day's schedule is:

• 10:00 AM Registration, Bagels and Coffee
• 11:00 AM Talk: Sara Billey
• 12:00 PM Lunch @ Poquitos
•   2:30 PM Talk: Shahriar Shahriari
•   3:00 PM Cookies, Coffee and Cokes
•   3:30 PM Talk: Marni Mishna
•   5:00 PM Happy Hour, Dinner

### Enumeration of Parabolic Double Cosets for Symmetric Groups and Beyond

Parabolic subgroups $W_I$ of Coxeter systems $(W,S)$ and their ordinary and double cosets $W / W_I$ and $W_I \backslash W / W_J$ appear in many contexts in combinatorics and Lie theory, including the geometry and topology of generalized flag varieties and the symmetry groups of regular polytopes. The set of ordinary cosets $w W_I$, for $I \subseteq S$, forms the Coxeter complex of $W$, and is well-studied. In this talk, we look at a less studied object: the set of all parabolic double cosets $W_I w W_J$ for $I, J \subseteq S$.

Each double coset can be presented by many different triples $(I,w,J)$. We describe what we call the lex-minimal presentation and prove that there exists a unique such choice for each double coset. Lex-minimal presentations can be enumerated via a finite automaton depending on the Coxeter graph for $(W,S)$.

In particular, we present a formula for the number of parabolic double cosets with a fixed minimal element when $W$ is the symmetric group $S_n$. In that case, parabolic subgroups are also known as Young subgroups. Our formula is almost always linear time computable in $n$, and the formula can be generalized to any Coxeter group.

This is talk is based on joint work with Matjaz Konvalinka, T. Kyle Petersen, William Slofstra and Bridget Tenner.

### Forbidden Configurations and other Combinatorial Problems for Posets of Subspaces

Let $V$ be a finite dimensional vector space over a finite field, and consider the poset of subspaces of $V$ ordered by inclusion. The combinatorial properties of this partially ordered set often resemble those of the Boolean lattices, the subsets of a finite set ordered by inclusion. Often the case of subsets is easier to handle, but, surprisingly, there are situations where a combinatorial question is easier to answer for subspaces. We survey a number of problems of each type including forbidden configuration problems: Given a small poset $P$, what is the largest number of subspaces of $V$ that do not contain a copy of $P$?

### The Remarkable Ubiquity of Standard Young Tableaux of Bounded Height

Standard Young tableaux are a classic family of combinatorial objects that appeared in algebra early in the previous century. Their utility is widely appreciated. The subfamily of tableaux with bounded height also appears in many guises in bijective and enumerative combinatorics. The generating functions are particularly lovely for their algebraic and analytic properties. We will explore these combinatorial classes, focussing on recent bijections that illustrate new, non-trivial connections between some very classic objects. We conclude by tracing the shadows of these results in representation theory.

Work in collaboration with Julien Courtiel, Eric Fusy and Mathias Lepoutre.

### Registration

The Combinatorial Potlatch has no permanent organization and no budget. And we like it that way. Consequently, there are no registration fees because we wouldn't know what to do with them. You are on your own for meals and lodging, and the sponsoring institutions provides facilities, food for the breaks and some support for speakers' travel. So expressions of appreciation to the speakers and the hosts are preferred and especially encouraged. Thanks.

### Getting There

All talks will be held in Casey 500, with registration and breaks nearby. Search with Casey on the interactive campus map, or locate in D-1 grid on the PDF campus map (which has North to the right of the page).

Driving directions at the Seattle University site.

Parking \$6/day on a Saturday. The P5 visitor lot, just inside the entrance at 12th and Marion, is probably the most convenient to the meeting location. There is an admissions event at Seattle U on the same day, so if parking is crowded or full, you can consult Diamond Parking or U Park. Seattle U page with details on visitor parking. Seattle University is about one mile from downtown Seattle, so can be reached from there by walking, taking the #12 Metro bus to Broadway & Madison, or a short taxi ride. ### Lodging The Silver Cloud Broadway Hotel is directly across the street from Seattle University. You can telephone 206-325-1400 and mention the Potlatch conference or the POTLATCH group for best available rate. You can also use this group link . Be sure you get the discounted rates: \$149/king room and \$159/double queen room. Parking is \$25/night. [More hotel information]

Seattle University is a mile from downtown Seattle along Madison Street. So any hotel downtown would also be an option.

### Dining and Happy Hour

You are encouraged to join other conference participants at the various meals and other events we are planning for the day. Details here as we have them.

Lunch: Poquitos [1000 E Pike St], 1/4 mile walking

Happy Hour: Barca [1510 11th Ave], 1/4 mile walking

Dinner: Stout [1530 11th Ave], 1/4 mile walking

### Organizers

• Rob Beezer, University of Puget Sound, beezer (at) ups (dot) edu, Communications Chair
• Nancy Ann Neudauer, Pacific University, nancy (at) pacificu (dot) edu, Program Chair
• Mark MacLean, Seattle University, macleanm (a) seattleu (dot) edu, Local Arrangements Chair
Last updated: November 18, 2016, http://buzzard.ups.edu/potlatch/2016/potlatch2016.html