7,377 research outputs found
On the number of fully packed loop configurations with a fixed associated matching
We show that the number of fully packed loop configurations corresponding to
a matching with nested arches is polynomial in if is large enough,
thus essentially proving two conjectures by Zuber [Electronic J. Combin. 11
(2004), Article #R13].Comment: AnS-LaTeX, 43 pages; Journal versio
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
Lattice paths of slope 2/5
We analyze some enumerative and asymptotic properties of Dyck paths under a
line of slope 2/5.This answers to Knuth's problem \\#4 from his "Flajolet
lecture" during the conference "Analysis of Algorithms" (AofA'2014) in Paris in
June 2014.Our approach relies on the work of Banderier and Flajolet for
asymptotics and enumeration of directed lattice paths. A key ingredient in the
proof is the generalization of an old trick of Knuth himself (for enumerating
permutations sortable by a stack),promoted by Flajolet and others as the
"kernel method". All the corresponding generating functions are algebraic,and
they offer some new combinatorial identities, which can be also tackled in the
A=B spirit of Wilf--Zeilberger--Petkov{\v s}ek.We show how to obtain similar
results for other slopes than 2/5, an interesting case being e.g. Dyck paths
below the slope 2/3, which corresponds to the so called Duchon's club model.Comment: Robert Sedgewick and Mark Daniel Ward. Analytic Algorithmics and
Combinatorics (ANALCO)2015, Jan 2015, San Diego, United States. SIAM, 2015
Proceedings of the Twelfth Workshop on Analytic Algorithmics and
Combinatorics (ANALCO), eISBN 978-1-61197-376-1, pp.105-113, 2015, 2015
Proceedings of the Twelfth Workshop on Analytic Algorithmics and
Combinatorics (ANALCO
Moderate Growth Time Series for Dynamic Combinatorics Modelisation
Here, we present a family of time series with a simple growth constraint.
This family can be the basis of a model to apply to emerging computation in
business and micro-economy where global functions can be expressed from local
rules. We explicit a double statistics on these series which allows to
establish a one-to-one correspondence between three other ballot-like
strunctures
Stammering tableaux
The PASEP (Partially Asymmetric Simple Exclusion Process) is a probabilistic
model of moving particles, which is of great interest in combinatorics, since
it appeared that its partition function counts some tableaux. These tableaux
have several variants such as permutations tableaux, alternative tableaux,
tree- like tableaux, Dyck tableaux, etc. We introduce in this context certain
excursions in Young's lattice, that we call stammering tableaux (by analogy
with oscillating tableaux, vacillating tableaux, hesitating tableaux). Some
natural bijections make a link with rook placements in a double staircase,
chains of Dyck paths obtained by successive addition of ribbons, Laguerre
histories, Dyck tableaux, etc.Comment: Clarification and better exposition thanks reviewer's report
The Matrix Ansatz, Orthogonal Polynomials, and Permutations
In this paper we outline a Matrix Ansatz approach to some problems of
combinatorial enumeration. The idea is that many interesting quantities can be
expressed in terms of products of matrices, where the matrices obey certain
relations. We illustrate this approach with applications to moments of
orthogonal polynomials, permutations, signed permutations, and tableaux.Comment: to appear in Advances in Applied Mathematics, special issue for
Dennis Stanto
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