148 research outputs found
The enumeration of generalized Tamari intervals
Let be a grid path made of north and east steps. The lattice
, based on all grid paths weakly above and
sharing the same endpoints as , was introduced by Pr\'eville-Ratelle and
Viennot (2014) and corresponds to the usual Tamari lattice in the case
. Our main contribution is that the enumeration of intervals in
, over all of length , is given by . This formula was first obtained by Tutte(1963) for
the enumeration of non-separable planar maps. Moreover, we give an explicit
bijection from these intervals in to non-separable
planar maps.Comment: 19 pages, 11 figures. Title changed, originally titled "From
generalized Tamari intervals to non-separable planar maps (extended
abstract)", submitte
An extension of Tamari lattices
For any finite path on the square grid consisting of north and east unit
steps, starting at (0,0), we construct a poset Tam that consists of all
the paths weakly above with the same number of north and east steps as .
For particular choices of , we recover the traditional Tamari lattice and
the -Tamari lattice.
Let be the path obtained from by reading the unit
steps of in reverse order, replacing the east steps by north steps and vice
versa. We show that the poset Tam is isomorphic to the dual of the poset
Tam. We do so by showing bijectively that the poset
Tam is isomorphic to the poset based on rotation of full binary trees with
the fixed canopy , from which the duality follows easily. This also shows
that Tam is a lattice for any path . We also obtain as a corollary of
this bijection that the usual Tamari lattice, based on Dyck paths of height
, is a partition of the (smaller) lattices Tam, where the are all
the paths on the square grid that consist of unit steps.
We explain possible connections between the poset Tam and (the
combinatorics of) the generalized diagonal coinvariant spaces of the symmetric
group.Comment: 18 page
Combinatorics of r-Dyck paths, r-Parking functions, and the r-Tamari lattices
This paper's aim is to present recent combinatorial considerations on r-Dyck
paths, r-Parking functions, and the r-Tamari lattices. Giving a better
understanding of the combinatorics of these objects has become important in
view of their (conjectural) role in the description of the graded character of
the Sn-modules of bivariate and trivariate diagonal coinvariant spaces for the
symmetric group.Comment: 36 pages, 12 figure
The representation of the symmetric group on m-Tamari intervals
An m-ballot path of size n is a path on the square grid consisting of north
and east unit steps, starting at (0,0), ending at (mn,n), and never going below
the line {x=my}. The set of these paths can be equipped with a lattice
structure, called the m-Tamari lattice and denoted by T_n^{m}, which
generalizes the usual Tamari lattice T_n obtained when m=1. This lattice was
introduced by F. Bergeron in connection with the study of diagonal coinvariant
spaces in three sets of n variables. The representation of the symmetric group
S_n on these spaces is conjectured to be closely related to the natural
representation of S_n on (labelled) intervals of the m-Tamari lattice, which we
study in this paper. An interval [P,Q] of T_n^{m} is labelled if the north
steps of Q are labelled from 1 to n in such a way the labels increase along any
sequence of consecutive north steps. The symmetric group S_n acts on labelled
intervals of T_n^{m} by permutation of the labels. We prove an explicit
formula, conjectured by F. Bergeron and the third author, for the character of
the associated representation of S_n. In particular, the dimension of the
representation, that is, the number of labelled m-Tamari intervals of size n,
is found to be (m+1)^n(mn+1)^{n-2}. These results are new, even when m=1. The
form of these numbers suggests a connection with parking functions, but our
proof is not bijective. The starting point is a recursive description of
m-Tamari intervals. It yields an equation for an associated generating
function, which is a refined version of the Frobenius series of the
representation. This equation involves two additional variables x and y, a
derivative with respect to y and iterated divided differences with respect to
x. The hardest part of the proof consists in solving it, and we develop
original techniques to do so, partly inspired by previous work on polynomial
equations with "catalytic" variables.Comment: 29 pages --- This paper subsumes the research report arXiv:1109.2398,
which will not be submitted to any journa
Counting smaller elements in the Tamari and m-Tamari lattices
We introduce new combinatorial objects, the interval- posets, that encode
intervals of the Tamari lattice. We then find a combinatorial interpretation of
the bilinear operator that appears in the functional equation of Tamari
intervals described by Chapoton. Thus, we retrieve this functional equation and
prove that the polynomial recursively computed from the bilinear operator on
each tree T counts the number of trees smaller than T in the Tamari order. Then
we show that a similar m + 1-linear operator is also used in the functionnal
equation of m-Tamari intervals. We explain how the m-Tamari lattices can be
interpreted in terms of m+1-ary trees or a certain class of binary trees. We
then use the interval-posets to recover the functional equation of m-Tamari
intervals and to prove a generalized formula that counts the number of elements
smaller than or equal to a given tree in the m-Tamari lattice.Comment: 46 pages + 3 pages of code appendix, 27 figures. Long version of
arXiv:1212.0751. To appear in Journal of Combinatorial Theory, Series
Geometric realizations of Tamari interval lattices via cubic coordinates
We introduce cubic coordinates, which are integer words encoding intervals in
the Tamari lattices. Cubic coordinates are in bijection with interval-posets,
themselves known to be in bijection with Tamari intervals. We show that in each
degree the set of cubic coordinates forms a lattice, isomorphic to the lattice
of Tamari intervals. Geometric realizations are naturally obtained by placing
cubic coordinates in space, highlighting some of their properties. We consider
the cellular structure of these realizations. Finally, we show that the poset
of cubic coordinates is shellable
Two bijections on Tamari intervals
We use a recently introduced combinatorial object, the interval-poset, to
describe two bijections on intervals of the Tamari lattice. Both bijections
give a combinatorial proof of some previously known results. The first one is
an inner bijection between Tamari intervals that exchanges the initial rise and
lower contacts statistics. Those were introduced by Bousquet-M\'elou, Fusy, and
Pr\'eville-Ratelle who proved they were symmetrically distributed but had no
combinatorial explanation. The second bijection sends a Tamari interval to a
closed flow of an ordered forest. These combinatorial objects were studied by
Chapoton in the context of the Pre-Lie operad and the connection with the
Tamari order was still unclear.Comment: 12 pages, 10 figure
Motzkin Intervals and Valid Hook Configurations
We define a new natural partial order on Motzkin paths that serves as an
intermediate step between two previously-studied partial orders. We provide a
bijection between valid hook configurations of -avoiding permutations and
intervals in these new posets. We also show that valid hook configurations of
permutations avoiding (or equivalently, ) are counted by the same
numbers that count intervals in the Motzkin-Tamari posets that Fang recently
introduced, and we give an asymptotic formula for these numbers. We then
proceed to enumerate valid hook configurations of permutations avoiding other
collections of patterns. We also provide enumerative conjectures, one of which
links valid hook configurations of -avoiding permutations, intervals in
the new posets we have defined, and certain closed lattice walks with small
steps that are confined to a quarter plane.Comment: 22 pages, 8 figure
Combinatorics of Labelled Parallelogram polyominoes
We obtain explicit formulas for the enumeration of labelled parallelogram
polyominoes. These are the polyominoes that are bounded, above and below, by
north-east lattice paths going from the origin to a point (k,n). The numbers
from 1 and n (the labels) are bijectively attached to the north steps of
the above-bounding path, with the condition that they appear in increasing
values along consecutive north steps. We calculate the Frobenius characteristic
of the action of the symmetric group S_n on these labels. All these enumeration
results are refined to take into account the area of these polyominoes. We make
a connection between our enumeration results and the theory of operators for
which the intergral Macdonald polynomials are joint eigenfunctions. We also
explain how these same polyominoes can be used to explicitly construct a linear
basis of a ring of SL_2-invariants.Comment: 25 pages, 9 figure
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