39 research outputs found
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
Riffle shuffles with biased cuts
The well-known Gilbert-Shannon-Reeds model for riffle shuffles assumes that
the cards are initially cut 'about in half' and then riffled together. We
analyze a natural variant where the initial cut is biased. Extending results of
Fulman (1998), we show a sharp cutoff in separation and L-infinity distances.
This analysis is possible due to the close connection between shuffling and
quasisymmetric functions along with some complex analysis of a generating
function.Comment: 10 page
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.
Card-Shuffling via Convolutions of Projections on Combinatorial Hopf Algebras
Recently, Diaconis, Ram and I created Markov chains out of the
coproduct-then-product operator on combinatorial Hopf algebras. These chains
model the breaking and recombining of combinatorial objects. Our motivating
example was the riffle-shuffling of a deck of cards, for which this Hopf
algebra connection allowed explicit computation of all the eigenfunctions. The
present note replaces in this construction the coproduct-then-product map with
convolutions of projections to the graded subspaces, effectively allowing us to
dictate the distribution of sizes of the pieces in the breaking step of the
previous chains. An important example is removing one "vertex" and reattaching
it, in analogy with top-to-random shuffling. This larger family of Markov
chains all admit analysis by Hopf-algebraic techniques. There are simple
combinatorial expressions for their stationary distributions and for their
eigenvalues and multiplicities and, in some cases, the eigenfunctions are also
calculable.Comment: 12 pages. This is an extended abstract, to appear in Proceedings of
the 27th International Conference on Formal Power Series and Algebraic
Combinatorics (FPSAC). Comments are very welcom