24,683 research outputs found
Permutads
We unravel the algebraic structure which controls the various ways of
computing the word ((xy)(zt)) and its siblings. We show that it gives rise to a
new type of operads, that we call permutads. It turns out that this notion is
equivalent to the notion of "shuffle algebra" introduced by the second author.
It is also very close to the notion of "shuffle operad" introduced by V.
Dotsenko and A. Khoroshkin. It can be seen as a noncommutative version of the
notion of nonsymmetric operads. We show that the role of the associahedron in
the theory of operads is played by the permutohedron in the theory of
permutads.Comment: Same results, re-arranged and more details. 38 page
Partial mixing of semi-random transposition shuffles
We show that for any semi-random transposition shuffle on cards, the
mixing time of any given cards is at most , provided . In the case of the top-to-random transposition shuffle we show that
there is cutoff at this time with a window of size O(n), provided further that
as (and no cutoff otherwise). For the
random-to-random transposition shuffle we show cutoff at time
for the same conditions on . Finally, we analyse the cyclic-to-random
transposition shuffle and show partial mixing occurs at time for some just larger than 1/2. We prove these results by relating
the mixing time of cards to the mixing of one card. Our results rely
heavily on coupling arguments to bound the total variation distance.Comment: 23 pages, 4 figure
Partial Derivative Automaton for Regular Expressions with Shuffle
We generalize the partial derivative automaton to regular expressions with
shuffle and study its size in the worst and in the average case. The number of
states of the partial derivative automata is in the worst case at most 2^m,
where m is the number of letters in the expression, while asymptotically and on
average it is no more than (4/3)^m
Operads with compatible CL-shellable partition posets admit a Poincar\'e-Birkhoff-Witt basis
In 2007, Vallette built a bridge across posets and operads by proving that an
operad is Koszul if and only if the associated partition posets are
Cohen-Macaulay. Both notions of being Koszul and being Cohen-Macaulay admit
different refinements: our goal here is to link two of these refinements. We
more precisely prove that any (basic-set) operad whose associated posets admit
isomorphism-compatible CL-shellings admits a Poincar\'e-Birkhoff-Witt basis.
Furthermore, we give counter-examples to the converse
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