1,993 research outputs found
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
Hopf algebras and Markov chains: Two examples and a theory
The operation of squaring (coproduct followed by product) in a combinatorial
Hopf algebra is shown to induce a Markov chain in natural bases. Chains
constructed in this way include widely studied methods of card shuffling, a
natural "rock-breaking" process, and Markov chains on simplicial complexes.
Many of these chains can be explictly diagonalized using the primitive elements
of the algebra and the combinatorics of the free Lie algebra. For card
shuffling, this gives an explicit description of the eigenvectors. For
rock-breaking, an explicit description of the quasi-stationary distribution and
sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes
will only appear on the version on Amy Pang's website, the arXiv version will
not be updated.
Combinatorics of balanced carries
We study the combinatorics of addition using balanced digits, deriving an
analog of Holte's "amazing matrix" for carries in usual addition. The
eigenvalues of this matrix for base b balanced addition of n numbers are found
to be 1,1/b,...,1/b^n, and formulas are given for its left and right
eigenvectors. It is shown that the left eigenvectors can be identified with
hyperoctahedral Foulkes characters, and that the right eigenvectors can be
identified with hyperoctahedral Eulerian idempotents. We also examine the
carries that occur when a column of balanced digits is added, showing this
process to be determinantal. The transfer matrix method and a serendipitous
diagonalization are used to study this determinantal process.Comment: 18 page
Invariant Theory for the free left-regular band and a q-analogue
We examine from an invariant theory viewpoint the monoid algebras for two
monoids having large symmetry groups. The first monoid is the free left-regular
band on letters, defined on the set of all injective words, that is, the
words with at most one occurrence of each letter. This monoid carries the
action of the symmetric group. The second monoid is one of its -analogues,
considered by K. Brown, carrying an action of the finite general linear group.
In both cases, we show that the invariant subalgebras are semisimple
commutative algebras, and characterize them using Stirling and -Stirling
numbers.
We then use results from the theory of random walks and random-to-top
shuffling to decompose the entire monoid algebra into irreducibles,
simultaneously as a module over the invariant ring and as a group
representation. Our irreducible decompositions are described in terms of
derangement symmetric functions introduced by D\'esarm\'enien and Wachs.Comment: final version, to appear in the Pacific Journal of Mathematic
A rule of thumb for riffle shuffling
We study how many riffle shuffles are required to mix n cards if only certain
features of the deck are of interest, e.g. suits disregarded or only the colors
of interest. For these features, the number of shuffles drops from 3/2 log_2(n)
to log_2(n). We derive closed formulae and an asymptotic `rule of thumb'
formula which is remarkably accurate.Comment: 27 pages, 5 table
Noncommutative Symmetric Functions and an Amazing Matrix
We present a simple way to derive the results of Diaconis and Fulman
[arXiv:1102.5159] in terms of noncommutative symmetric functions.Comment: 6 page
Carries, Shuffling and An Amazing Matrix
The number of ``carries'' when random integers are added forms a Markov
chain [23]. We show that this Markov chain has the same transition matrix as
the descent process when a deck of cards is repeatedly riffle shuffled.
This gives new results for the statistics of carries and shuffling.Comment: 16 page
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