*Date:* Jan. 28 , 2021, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:* Victor S. Miller, Center for Communications Research, Princeton.

*Title*: Counting Matrices that are Squares

*Abstract*:
On the math-fun mailing list (7 May 2013), Neil Sloane asked
to calculate the number of n times n matrices with entries in
{0,1} which are squares of other such matrices. In this paper we
analyze the case that the arithmetic is in GF(2) We follow
the dictum of Wilf (in the paper ``What is an answer?'') to derive a
``effective'' algorithm to count such matrices in much less time than
it takes to enumerate them. The algorithm which we use involves the
analysis of conjugacy classes of matrices. The restricted integer
partitions which arise are counted by the coefficients of one of
Ramanujan's mock Theta functions, which we found thanks to Sloane's
OEIS (Online Encyclopedia of Integer Sequences).

*Title*: Cyclic permutations, shuffles, and quasi-symmetric functions

*Abstract*: Richard Stanley proved that the distribution of descent number over all the shuffles of two permutations depends only on the descent
numbers of the permutations.

We present an explicit cyclic analogue of this result. The tools used for the proof include a new cyclic counterpart of Gessels quasi-symmetric functions, an unusual ring homomorphism, and a mysterious binomial identity with an interesting history.

Based on joint work with Ira Gessel, Vic Reiner, and Yuval Roichman.

*Date:* Feb. 11 , 2021, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:* Colin Defant, Princeton University

*Title*: Cumulants and Stack-Sorting

*Abstract*: Cumulant sequences are numerical sequences that play a fundamental role in noncommutative probability theory. West's stack-sorting map is a
combinatorially-defined operator that acts on permutations. In this talk, we will discuss how cumulants and stack-sorting, two topics from very disparate worlds,
are actually very closely related. This unexpected connection allows us to use tools from noncommutative probability theory to prove difficult,
surprising, and (occasionally) weird facts about the stack-sorting map. We will explore several applications of this method.

*Title*: Game Complexity: Between Geography and Santorini

*Abstract*: Santorini, a board game designed by mathematician Gordon Hamilton, is a two-player game of perfect information (a partizan combinatorial game) with simple rules and great depth of strategy. The structure of Santorini suggests a number of possible generalizations and simplifications that can be studied. We are interested in the difficulty of finding winning strategies in this family of games. One of our variants of Santorini turns out to be equivalent to a classical game called Geography, which is known to be "hard." In this talk, we present some background on Santorini, Geography, and game complexity, and we discuss what we can say about certain variants of Santorini.

Based on joint work with Carson Geissler.

*Date:* Feb. 25 , 2021, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:*
Aaron Robertson, Colgate University

*Title*: Data Mining and Ramsey Theory

*Abstract*:
Ramsey theory concerns itself with the emergence of patterns in sufficiently large structures.
Data miners search for patterns in extremely large data sets.
This is a cautionary tale for data miners about making inferences from large data
sets wherein we provide a framework for testing whether or not a correlation is spurious.

*Date:* March 4 , 2021, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:* David Nacin, William Patterson University.

*Title*: Padovan, Pascal, and Proofs without Words

*Abstract*:
What happens when we attempt to construct the Fibonacci spiral with triangles instead of squares? We get a new sequence, the Padovan sequence, which answers its own collection of unique and beautiful counting problems. In this talk we will show how this construction defines this sequence and then rediscover the same sequence hiding again in other surprising places. We then prove several identities without using either words or numbers, by considering triangles composed of colored dots.

The Fibonacci sequence is connected to the golden ratio which arises from a simple question about rectangles and proportion. A slightly different natural question leads to a new ratio and yet another method for defining our sequence. We then observe the uses of this sequence and its ratio in architecture and discuss the history behind the patterns we've uncovered. We conclude with a counterexample to a conjecture about this sequence that leads us to a final construction involving copies of the Fibonacci sequence itself.

*Date:* March 11 , 2021, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:* Michael Kiessling, Rutgers University

*Title*: A Maple-assisted study of a Schroedinger-Newton, a.k.a. Schroedinger-Poisson, a.k.a. Choquard, a.k.a. Pekar, a.k.a. ... equation

*Abstract*: In several different contexts (mathematical) physicists have proposed a nonlinear system of PDEs which can be recast
into a single Schroedinger equation with a Schroedinger potential that is the solution to a Poisson equation with the square
of the solution of the Schroedinger equation as source term; as such it is known under a variety of names (see above). For
instance, Roger Penrose proposed it in his theory of gravity-induced quantum-mechanical wave function collapse, but also
down-to-earth condensed matter theories, by Pekar and later by Choquard, produce this equation without invoking gravity.
Interestingly, the question of the asymptotic large-distance behavior has received several plausible but conflicting answers
which cannot all be true simultaneously. Recently I managed to settle the issue of the leading order term --- partly rigorously,
partly with the help of Maple. Subsequently Andrey Yudin from Moskow managed to tickle Maple to produce an intriguing
formula for all the putative asymptotic correction terms to the leading order term which are of power law type. Numerically
this formula seems pretty accurate, but it has yet to be proved; moreover, there are correction terms beyond all orders of
powers, and a formula for these terms has yet to be found

*Date:* March 25 , 2021, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:*
Alejandro Morales, UMass, Amherst, and William Shi, Northview highschool, Johns Creek, GA.
[starting Harvard, Fall 2021]

*Title*: Refinements and symmetries for volumes of flow polytopes

*Abstract*: Flow polytopes are an important class of polytopes in combinatorics whose lattice points and volumes have interesting properties and relations. The Chan-Robbins-Yuen (CRY) polytope is a flow polytope with normalized volume equal to the product of consecutive Catalan numbers. Zeilberger proved this by evaluating the Morris constant term identity, but no combinatorial proof is known. There is a refinement of this formula that splits the largest Catalan number into Narayana numbers, which Mészáros gave an interpretation as the volume of a collection of flow polytopes. We introduce a new refinement of the Morris identity with combinatorial interpretations both in terms of lattice points and volumes of flow polytopes. Our results generalize Mészáros's construction and a recent flow polytope interpretation of the Morris identity by Corteel-Kim-Mészáros. We prove the product formula of our refinement following the strategy of the Baldoni-Vergne proof of the Morris identity.

slides (w/o annotations) slides (w annotations)

*Date:* April 1 , 2021, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:* Doron Zeilberger, Rutgers University

*Title*: Proofs of the Riemann Hypothesis, the P ≠ NP conjecture, the Goldbach Cojecture, and the Irrationality of γ

*Abstract*: We present proofs of these conjectures, and time permitting, a few other ones.

[Joint work with Shalosh B. Ekhad]

*Date:* April 8 , 2021, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:*
Eric Sundberg, Occidental College

*Title*: Pairing Strategies for Tic-Tac-Toe on the Boolean Hypercube

*Abstract*: We consider a tic-tac-toe-style game on the vertices of the n-dimensional Boolean hypercube
{0,1}^{n} with k-dimensional subcubes as winning sets. We describe a pairing strategy which allows the second player to force
a draw when k = n/4 +1 in the case where n is a power of 4. Our results arose from significant experimentation using Mathematica.

(Based on joint work with Klay Kruczek and Ramin Naimi)

*Date:* April 15 , 2021, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:* Igor Pak, UCLA.

*Title*: What is a combinatorial interpretation

*Abstract*: The question in the title is deceptively simple, as the answers tend to be the number of certain trees, lattice paths, Young tableaux, and other friendly combinatorial objects.
However, the question lies in the heart of connections between enumerative/algebraic combinatorics and computer science.
I will survey what is known about the subject, and discuss some of my recent results.

*Date:* April 22 , 2021, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:* Toufik Mansour, University of Haifa

*Title*: On Stanley-Wilf limit of the pattern 1324

*Abstract*: We present an explicit formula for the generating function for the number of permutations of length $n$ that avoid
1324 in terms of generating functions for permutations that have a kernel shape of length m,
m ≥2. This allows us to write down a systematic procedure for finding a lower bound for approximating the Stanley-Wilf limit of the pattern 1324.

Joint work with Christian Nassau

*Date:* April 29 , 2021, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:* Victor S. Miller, IDA Center for Communications Research, Princeton

*Title*: Locality preserving hash functions, a partial order and tiles
in binary space

*Abstract*: A "tile" in the space B^n of bit vectors of length n, is a
subset S of B^n, such that there is another subset A of B^n so that
every element of B^n can be written uniquely in the form a + s, where
a in A and s in S. A particular class of tiles are the subsets of
minimum weight elements in the cosets of a linear code over GF(2). In
systematically investigating locality preserving hash functions, we
generated the list of all possible tiles of cardinality <=64
satisfying a certain optimality condition. All but 10 of them turned
out to be the sets of minimum weight elements described above.
Attempts to prove the same for the remaining 10 remained elusive.
Instead we found two computational criteria -- one using linear
programming, and the other using combinatorial bin packing, which
showed that remaining 10 could not be tiles.

[Joint with Don Coppersmith, Dan Gordon and Peter Ostapenko]

*Date:* May 6 , 2021, 5:00pm (Eastern Time)
Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

*Speaker:*
Adi Ben Israel, Rutgers University (RUTCOR).

*Title*: Data analysis in high-dimensional spaces

*Abstract*:

1. The unreliability of the Euclidean distance in high-dimension, making a proximity query meaningless and unstable because there is poor discrimination between the nearest and furthest neighbor [3], see also [4].

2. The uniform probability distribution on the n-dimensional unit sphere S_n, and some non-intuitive results for large n. For example, if x is any point in S_n, taken as the "north pole", then most of the area of S_n is concentrated in the "equator".

3. The advantage of the ℓ_{1}-distance, which is less sensitive to high dimensionality, and has been shown to
"provide the best discrimination in high-dimensional data spaces," [1, p. 427].

4. Clustering high-dimensional data using the ℓ_{1} distance, [2].

References

[1] C.C. Aggarwal et al, On the surprising behavior of distance metrics in high dimensional space, Lecture Notes in Computer Science, vol 1973(2001), Springer, https://doi.org/10.1007/3-540-44503-X_27

[2] T. Asamov and A. Ben-Israel, A probabilistic ℓ_{1} method for clustering high-dimensional data,
Probability in the Engineering and Informational Sciences, 2021, 1-16

[3] K. Beyer et al, When is "nearest neighbor" meaningful?, Lecture Notes in Computer Science, vol 1540(1999), Springer, https://doi.org/10.1007/3-540-49257-7_15

[4] J.M. Hammersley, The distribution of distance in a hypersphere, The Annals of Mathematical Statistics 21(1950), 447-452.

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