155 research outputs found
A permutation code preserving a double Eulerian bistatistic
Visontai conjectured in 2013 that the joint distribution of ascent and
distinct nonzero value numbers on the set of subexcedant sequences is the same
as that of descent and inverse descent numbers on the set of permutations. This
conjecture has been proved by Aas in 2014, and the generating function of the
corresponding bistatistics is the double Eulerian polynomial. Among the
techniques used by Aas are the M\"obius inversion formula and isomorphism of
labeled rooted trees. In this paper we define a permutation code (that is, a
bijection between permutations and subexcedant sequences) and show the more
general result that two -tuples of set-valued statistics on the set of
permutations and on the set of subexcedant sequences, respectively, are
equidistributed. In particular, these results give a bijective proof of
Visontai's conjecture
Avoiding patterns in irreducible permutations
International audienceWe explore the classical pattern avoidance question in the case of irreducible permutations, i.e., those in which there is no index such that . The problem is addressed completely in the case of avoiding one or two patterns of length three, and several well known sequences are encountered in the process, such as Catalan, Motzkin, Fibonacci, Tribonacci, Padovan and Binary numbers. Also, we present constructive bijections between the set of Motzkin paths of length and the sets of irreducible permutations of length (respectively fixed point free irreducible involutions of length ) avoiding a pattern for . This induces two new bijections between the set of Dyck paths and some restricted sets of permutations
A lattice on Dyck paths close to the Tamari lattice
We introduce a new poset structure on Dyck paths where the covering relation
is a particular case of the relation inducing the Tamari lattice. We prove that
the transitive closure of this relation endows Dyck paths with a lattice
structure. We provide a trivariate generating function counting the number of
Dyck paths with respect to the semilength, the numbers of outgoing and incoming
edges in the Hasse diagram. We deduce the numbers of coverings, meet and join
irreducible elements. As a byproduct, we present a new involution on Dyck paths
that transports the bistatistic of the numbers of outgoing and incoming edges
into its reverse. Finally, we give a generating function for the number of
intervals, and we compare this number with the number of intervals in the
Tamari lattice
Grand Dyck paths with air pockets
Grand Dyck paths with air pockets (GDAP) are a generalization of Dyck paths
with air pockets by allowing them to go below the -axis. We present
enumerative results on GDAP (or their prefixes) subject to various restrictions
such as maximal/minimal height, ordinate of the last point and particular first
return decomposition. In some special cases we give bijections with other known
combinatorial classes.Comment: 20 pages, 4 figure
More restrictive Gray codes for some classes of pattern avoiding permutations
International audienc
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