#Riffle mathematica full
#Riffle mathematica free

The main results concern the cycle structure of the resulting random permutations in Sn when n is large.

Next, 'riffle' the two stacks by dropping cards o.Ī class of models for riffle shuffles ('f Gammashuffles') related to certain expansive mappings of the unit interval is studied. For 0 ! p ! 1 and a deck of size N, the p-shuffle is defined as follows: First, 'cut' the deck into two stacks in such a way that the cardinality of the top stack has the Binomial (N p) distribution. Introduction The p-shuffle (or GSR-p shuffles, for Gilbert, Shannon, and Reeds) is a simple and natural generalization of the dovetail (or GSR) shuffle studied in and. The main result of the paper is an asymptotic (N !1) bound on the number of repetitions needed to 'randomize' the deck. Then, conditional on the outcome of the cut, the two stacks are 'riffled' in such a way that all possible riffles (interleavings) of these two stacks are equally likely. First, the deck is cut into a top stack and a bottom stack so that the distribution of the size of the top stack is Binomial (N p), where N is the total number of cards in the deck. The p-shuffle is a natural generalization of the dovetail shuffle that is defined as follows. This thesis will focus respectively on aįree-commutative basis, which produces 'independent breaking' chains, and aĬofree basis there will be both general statements and in-depth examples. One must restrict attention to certain styles of Hopf algebras in order to On the structure of the underlying Hopf algebra and its distinguished basis,

Since Hopf-power Markov chains can exhibit very different behaviour depending

#Riffle mathematica free

Theorem, Reutenauer's structure theory of the free Lie algebra, and Patras's This arises from aĬocktail of the Poincare-Birkhoff-Witt theorem, the Cartier-Milnor-Moore Underlying Hopf algebra is commutative or cocommutative.

#Riffle mathematica full

Perhaps the pinnacle is Theorem 2.5.1, a collection ofĪlgorithms for a full left and right eigenbasis in many common cases where the For example, there is an explicit formula for the stationaryĭistribution (Theorem 4.5.1), and constructing quotient algebras show thatĬertain statistics on a Hopf-power Markov chain are themselves Markov chains Them comes simply from translating well-known facts on the underlying HopfĪlgebra. Rephrasing familiar chains through this lens, is that much information about The primary reason for constructing Hopf-power Markov chains, or for Riffle-shuffling of a deck of cards, which arises in this manner from the Motivating example is the famous Gilbert-Shannon-Reeds model of These chains describe theīreaking-then-recombining of the combinatorial objects in the Hopf algebra. Matrix of the coproduct-then-product operator on a combinatorial Hopf algebra Transition matrix of a 'Hopf-power Markov chain' is (the transpose of) the This thesis introduces a way to build Markov chains out of Hopf algebras.