| Talk_id | Date | Speaker | Title |
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29100
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Wednesday 9/8
3:00 PM
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Emily Gunawan, Oklahoma University
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Box-ball systems and Robinson-Schensted-Knuth tableaux
- Emily Gunawan, Oklahoma University
- Box-ball systems and Robinson-Schensted-Knuth tableaux
- 09/08/2021
- 3:00 PM - 3:50 PM
- Online (virtual meeting)
(Virtual Meeting Link)
- Bruce E Sagan (bsagan@msu.edu)
The Robinson-Schensted (RS) correspondence is a famous bijection between permutations and pairs (P,Q) of standard tableaux of the same shape, called the RS partition. The RS partition and its conjugate record certain permutation statistics called Greene’s theorem statistics.
A box-ball system is a discrete dynamical system which can be thought of as a collection of time states. A permutation on n objects gives a box-ball system state by assigning its one-line notation to n consecutive boxes. After a finite number of steps, a box-ball system will reach a steady state. From any steady state, we can construct a tableau (not necessarily standard) called the soliton decomposition. The shape of the soliton decomposition is called the BBS partition. An exciting discovery (made in 2019 by Lewis, Lyu, Pylyavskyy, and Sen) is that the BBS partition and its conjugate record a localized version of Greene’s theorem statistics.
We will discuss a few new results:
(1) The Q tableau of a permutation completely determines the dynamics of the corresponding box-ball system.
(2) The permutations whose BBS partitions are L-shaped have steady-state time at most 1. This large class of permutations include column reading words and noncrossing involutions.
(3) If the soliton decomposition of a permutation is a standard tableau or if its BBS partition coincides with its RS partition, then its soliton decomposition and its P tableau are equal.
(4) Finally, we study the permutations whose P tableaux and soliton decompositions coincide and refer to them as “good". These “good” permutations are closed under consecutive pattern containment. Furthermore, we conjecture that the “good” Q tableaux are counted by the Motzkin numbers.
This talk is based on REU projects with Ben Drucker, Eli Garcia, Aubrey Rumbolt, Rose Silver (UConn Math REU 2020) and Marisa Cofie, Olivia Fugikawa, Madelyn Stewart, David Zeng (SUMRY 2021).
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29114
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Wednesday 9/15
3:00 PM
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Oliver Pechenik, University of Waterloo
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What is the degree of a Grothendieck polynomial?
- Oliver Pechenik, University of Waterloo
- What is the degree of a Grothendieck polynomial?
- 09/15/2021
- 3:00 PM - 3:50 PM
- Online (virtual meeting)
(Virtual Meeting Link)
- Bruce E Sagan (bsagan@msu.edu)
Jenna Rajchgot observed that the Castelnuovo-Mumford regularity of matrix Schubert varieties is computed by the degrees of the corresponding Grothendieck polynomials. We give a formula for these degrees. Indeed, we compute the leading terms of the top degree pieces of Grothendieck polynomials and give a complete description of when two Grothendieck polynomials have the same top degree piece (up to scalars). Our formulas rely on some new facts about major index of permutations. (Joint work with David Speyer and Anna Weigandt.)
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29123
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Wednesday 9/22
3:00 PM
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Einar Steingrímsson, University of Strathclyde
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Permutation statistics and moment sequences
- Einar Steingrímsson, University of Strathclyde
- Permutation statistics and moment sequences
- 09/22/2021
- 3:00 PM - 3:50 PM
- Online (virtual meeting)
(Virtual Meeting Link)
- Bruce E Sagan (bsagan@msu.edu)
Which combinatorial sequences correspond to moments of probability measures on the real line? We present a generating function, as a continued fraction, for a 14-parameter family of integer sequences and interpret these in terms of statistics on permutations and other combinatorial objects. Special cases include several classical and noncommutative probability laws, and a substantial subset of the orthogonalizing measures in the q-Askey scheme of orthogonal polynomials.
This continued fraction captures a variety of combinatorial sequences. In particular, it characterizes the moment sequences associated to the numbers of permutations avoiding (classical and vincular) patterns of length three. This connection between pattern avoidance and classical and noncommutative probability is among several consequences that generalize and unify previous results in the literature.
The fourteen combinatorial statistics further generalize to colored permutations, and, as an infinite family of statistics, to the k-arrangements: permutations with k-colored fixed points, introduced here. This is joint work with Natasha Blitvić, Lancaster University.
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29137
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Wednesday 10/6
3:00 PM
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Richard A. Brualdi, University of Wisconsin - Madison
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About Permutation Matrices
- Richard A. Brualdi, University of Wisconsin - Madison
- About Permutation Matrices
- 10/06/2021
- 3:00 PM - 3:50 PM
- Online (virtual meeting)
(Virtual Meeting Link)
- Bruce E Sagan (bsagan@msu.edu)
The study of permutations is both ancient and modern. They can be viewed as the integers $1,2,\ldots,n$ in some order or as $n\times n$ permutation matrices. They can be regarded as data which is to be sorted. The explicit definition of the determinant uses permutations. An inversion of a permutation occurs when a larger integer precedes a smaller integer. Inversions can be used to define two partial orders on permutations, one weaker than the other. Partial orders have a unique minimal completion to a lattice, the Dedekind-MacNeille completion. Generalizations of permutation matrices determine related matrix classes, for instance, alternating sign matrices (ASMs) which arose independently in the mathematics and physics literature. Permutations may contain certain patterns, e.g. three integers in increasing order; avoiding such patterns determines certain permutation classes. Similar restrictions can be placed more generally on $(0,1)$-matrices. The convex hull of $n\times n$ permutation matrices is the polytope of $n\times n$ doubly stochastic matrices. In a similar way we get ASM polytopes. We shall explore these and other ideas and their connections.
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29150
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Wednesday 10/13
3:00 PM
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Greta Panova, USC
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Hook length formulas for skew shapes and beyond
- Greta Panova, USC
- Hook length formulas for skew shapes and beyond
- 10/13/2021
- 3:00 PM - 3:50 PM
- Online (virtual meeting)
(Virtual Meeting Link)
- Bruce E Sagan (bsagan@msu.edu)
Abstract: The hook length formula for the number of Standard Young Tableaux of a partition lambda is one of the few miraculous product formulas we see in combinatorics. While no such formula for the number of skew SYTs exists in general, the recent Naruse Hook Length Formula (NHLF) brings us close. I will explain this formula and generalizations, give some proofs and bijections, and discuss new extensions of NHLF for increasing tableaux originating in the study of Grothendieck polynomials.
Based on a series of papers with A. Morales and I. Pak
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29155
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Wednesday 10/20
3:00 PM
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Jessica Striker, North Dakota State University
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Promotion, rotation, and a web basis of invariant polynomials from noncrossing partitions
- Jessica Striker, North Dakota State University
- Promotion, rotation, and a web basis of invariant polynomials from noncrossing partitions
- 10/20/2021
- 3:00 PM - 3:50 PM
- Online (virtual meeting)
(Virtual Meeting Link)
- Bruce E Sagan (bsagan@msu.edu)
Many combinatorial objects with strikingly good enumerative formulae and dynamical behavior (such as cyclic sieving) have underlying algebraic meaning. We first review classical results on promotion of standard Young tableaux, rotation of matchings/webs, and related invariant polynomials and symmetric group actions. We then discuss recent joint work with Rebecca Patrias and Oliver Pechenik involving the more general setting of increasing tableaux and noncrossing partitions.
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29172
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Wednesday 10/27
3:00 PM
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Quinn Minnich, Michigan State University
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Counting Admissible Orderings of a Pinnacle Set
- Quinn Minnich, Michigan State University
- Counting Admissible Orderings of a Pinnacle Set
- 10/27/2021
- 3:00 PM - 3:50 PM
- Online (virtual meeting)
(Virtual Meeting Link)
- Bruce E Sagan (bsagan@msu.edu)
Let $S_n$ be the symmetric group. The pinnacle set of a permutation in $S_n$ is defined to be all elements of that permutation which are larger than both of their adjacent elements. Given a subset $P$ of $\{1,\ldots,n\}$ we can also ask if there exists a permutation in $S_n$ having $P$ as its pinnacle set. If so, we say $P$ is admissible. We can extend this idea further by ordering the elements of $P$ and asking if there exists a permutation in $S_n$ having pinnacle set $P$ with the elements of $P$ in the given order. If so, we say that the ordering is an admissible ordering. In this presentation, we will present an efficient recursion for counting the number of admissible orderings of a given pinnacle set.
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