##### Department of Mathematics,

University of California San Diego

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### Math 269 - Combinatorics

## Todd Kemp

#### UCSD

## Non-Crossing Pair Partitions and Bit Strings

##### Abstract:

\indent The lattice of non-crossing partitions of $[n]$, $NC(n)$, is a highly-studied, highly-symmetric playground for both algebraic and enumerative combinatorics. First seriously studied by Kreweras (1972), it has more recently come to light in two very different contexts: as the combinatorial underpinnings of the new, fruitful field of Free Probability, and having new unexpected connections to Coxeter groups and hyperplane arrangements (both discoveries made in the last 15 years). \indent In this lecture, we will discuss non-crossing pair partitions of $[2n]$, $NC_2(2n)$. This set is in natural bijection with $NC(n)$. Consider the subset of those pair partitions that only pair $1$s to $0$s in some random bit-string of length $2n$. The enumerative properties of such classes of pairings are extremely important to some hard problems in free probability theory and random matrix theory. We will discuss what is known about this enumeration (including a "tight" theorem on symmetry and maximization), as well as some surprising and suggestive algebraic properties of posets associated to these pairings. \indent This is joint work with Mahrlburg, Rattan, and Smyth, as well as Chou, Fricano, Poh, Shore, Whieldon, Wong, and Zhang.

### May 24, 2011

### 4:00 PM

### AP&M 7321

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