53 research outputs found
Affine shuffles, shuffles with cuts, the Whitehouse module, and patience sorting
Type A affine shuffles are compared with riffle shuffles followed by a cut.
Although these probability measures on the symmetric group S_n are different,
they both satisfy a convolution property. Strong evidence is given that when
the underlying parameter satisfies , the induced measures on
conjugacy classes of the symmetric group coincide. This gives rise to
interesting combinatorics concerning the modular equidistribution by major
index of permutations in a given conjugacy class and with a given number of
cyclic descents. It is proved that the use of cuts does not speed up the
convergence rate of riffle shuffles to randomness. Generating functions for the
first pile size in patience sorting from decks with repeated values are
derived. This relates to random matrices.Comment: Galley version for J. Alg.; minor revisions in Sec.
The combinatorics of biased riffle shuffles
This paper studies biased riffle shuffles, first defined by Diaconis, Fill,
and Pitman. These shuffles generalize the well-studied Gilbert-Shannon-Reeds
shuffle and convolve nicely. An upper bound is given for the time for these
shuffles to converge to the uniform distribution; this matches lower bounds of
Lalley. A careful version of a bijection of Gessel leads to a generating
function for cycle structure after one of these shuffles and gives new results
about descents in random permutations. Results are also obtained about the
inversion and descent structure of a permutation after one of these shuffles.Comment: 11 page
Applications of the Brauer complex: card shuffling, permutation statistics, and dynamical systems
By algebraic group theory, there is a map from the semisimple conjugacy
classes of a finite group of Lie type to the conjugacy classes of the Weyl
group. Picking a semisimple class uniformly at random yields a probability
measure on conjugacy classes of the Weyl group. Using the Brauer complex, it is
proved that this measure agrees with a second measure on conjugacy classes of
the Weyl group induced by a construction of Cellini using the affine Weyl
group. Formulas for Cellini's measure in type are found. This leads to new
models of card shuffling and has interesting combinatorial and number theoretic
consequences. An analysis of type C gives another solution to a problem of
Rogers in dynamical systems: the enumeration of unimodal permutations by cycle
structure. The proof uses the factorization theory of palindromic polynomials
over finite fields. Contact is made with symmetric function theory.Comment: One change: we fix a typo in definition of f(m,k,i,d) on page 1
Carries, shuffling, and symmetric functions
The "carries" when n random numbers are added base b form a Markov chain with
an "amazing" transition matrix determined by Holte. This same Markov chain
occurs in following the number of descents or rising sequences when n cards are
repeatedly riffle shuffled. We give generating and symmetric function proofs
and determine the rate of convergence of this Markov chain to stationarity.
Similar results are given for type B shuffles. We also develop connections with
Gaussian autoregressive processes and the Veronese mapping of commutative
algebra.Comment: 23 page
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