# Descent-inversion statistics in riffle shuffles

@article{Islak2013DescentinversionSI, title={Descent-inversion statistics in riffle shuffles}, author={Umit Islak}, journal={Turkish Journal of Mathematics}, year={2013}, volume={42} }

This paper studies statistics of riffle shuffles by relating them to random word statistics with the use of inverse shuffles. Asymptotic normality of the number of descents and inversions in riffle shuffles with convergence rates of order $1/\sqrt{n}$ in the Kolmogorov distance are proven. Results are also given about the lengths of the longest alternating subsequences of random permutations resulting from riffle shuffles. A sketch of how the theory of multisets can be useful for statistics of… Expand

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#### References

SHOWING 1-10 OF 25 REFERENCES

The Combinatorics of Biased Riffle Shuffles

- Mathematics, Computer Science
- Comb.
- 1998

Biasful riffle shuffles generalize the well-studied Gilbert-Shannon-Reeds shuffle and convolve nicely, and an upper bound is given for the time for these shuffles to converge to the uniform distribution. Expand

Analysis of Top To Random Shuffles

- Computer Science, Mathematics
- Comb. Probab. Comput.
- 1992

A deck of n cards is shuffled by repeatedly taking off the top m cards and inserting them in random positions to give the asymptotics of the approach to stationarity: for m fixed and n large, it takes shuffles to get close to random. Expand

Longest alternating subsequences of k-ary words

- Computer Science, Mathematics
- Discret. Appl. Math.
- 2008

An explicit formula is found for the generating function of the number of k-ary words of length n according to the length of the longest alternating subsequence of a permutation in the symmetric group. Expand

Trailing the Dovetail Shuffle to its Lair

- Mathematics
- 1992

We analyze the most commonly used method for shuffling cards. The main result is a simple expression for the chance of any arrangement after any number of shuffles. This is used to give sharp bounds… Expand

A Probabilistic Approach to the Asymptotics of the Length of the Longest Alternating Subsequence

- Mathematics, Computer Science
- Electron. J. Comb.
- 2010

The methodology is robust enough to tackle similar problems for finite alphabet random words or even Markovian sequences, and a sketch of how some cases of pattern restricted permutations can also be tackled with probabilistic methods is presented. Expand

The number of flags in finite vector spaces: asymptotic normality and Mahonian statistics

- Mathematics
- 2013

We study the generalized Galois numbers which count flags of length r in N-dimensional vector spaces over finite fields. We prove that the coefficients of those polynomials are asymptotically… Expand

Local extrema in random permutations and the structure of longest alternating subsequences

- Mathematics
- 2011

Let asn denote the length of a longest alternating subsequence in a uniformly random permutation of order n. Stanley studied the distribution of asn using algebraic methods, and showed in particular… Expand

Unfair permutations

- Computer Science, Medicine
- Eur. J. Comb.
- 2011

We study unfair permutations, which are generated by letting n players draw numbers and assuming that player i draws i times from the unit interval and records her largest value. This model is… Expand

Convergence Rates for Generalized Descents

- Mathematics, Computer Science
- Electron. J. Comb.
- 2011

An explicit formula is provided for the mean and variance of d -descents of permutations of length n and bounds on the rate of convergence using Stein's method are obtained. Expand

Normal Approximations for Descents and Inversions of Permutations of Multisets

- Mathematics
- 2005

Abstract
Normal approximations for descents and inversions of permutations of the set {1,2,…,n} are well known. We consider the number of inversions of a permutation π(1),π(2),…,π(n) of a multiset… Expand