The identity is the most likely exchange shuffle for large n

A quadratic identity in the shuffle algebra and an alternative proof for de Bruijn's formula

Motivated by a polynomial identity of certain iterated integrals, first observed in [CGM20] in the setting of lattice paths, we prove an intriguing combinatorial identity in the shuffle algebra. It has a close connection to de Bruijn's formula when interpreted in the framework of signatures of paths.

A Shuffle Theorem for Paths Under Any Line

We generalize the shuffle theorem and its $(km,kn)$ version, as conjectured by Haglund et al. and Bergeron et al., and proven by Carlsson and Mellit, and Mellit, respectively. In our version the $(km,kn)$ Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whose $x$ and $y$ intercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra operator formula or an equivalent explicit raising operator formula. We derive our combinatorial identity as the polynomial truncation of an identity of infinite series of $GL_{l}$ charac…

What is the Perfect Shuffle?

When shuffling a deck of cards, one probably wants to make sure it is thoroughly shuffled. A way to do this is by sifting through the cards to ensure that no adjacent cards are the same number, because surely this is a poorly shuffled deck. Unfortunately, human intuition for probability tends to lead us astray. For a standard 52-card deck of playing cards, the event is actually extremely likely. This report will attempt to elucidate how to answer this surprisingly difficult combinatorial question directly using rook polynomials.

贝内什和混洗交换反例

Beneš and Shuffle-Exchange Counterexamples

本文给出混洗型网络可重排性猜想的反例。构造\(N\)正则图反驳贝内什不等式，证明标准有向混洗交换网络中\(d(k,3)=6(k\geq3)\)，说明混洗交换猜想不成立，建议用\(d(k,n)=3n - 3\)替代原猜想。

Riffle shuffles of decks with repeated cards

By a well-known result of Bayer and Diaconis, the maximum entropy model of the common riffle shuffle implies that the number of riffle shuffles necessary to mix a standard deck of 52 cards is either 7 or 11--with the former number applying when the metric used to define mixing is the total variation distance and the latter when it is the separation distance. This and other related results assume all 52 cards in the deck to be distinct and require all $52!$ permutations of the deck to be almost equally likely for the deck to be considered well mixed. In many instances, not all cards in the deck…

The weak Bruhat order for random walks on Coxeter groups

We show that for the simple random walk on a Coxeter group generated by the Coxeter generators and identity, the likelihoods of being at any pair of states respect the weak Bruhat order. That is, after any number of steps, the most likely element is the identity, probabilities decrease along any geodesic from the identity, and the least likely element is the longest element, if the group is finite. The result remains true when different generators have different probabilities, so long as the identity is at least as likely as any other.