3,247 research outputs found
Counting Arithmetical Structures on Paths and Cycles
Let G be a finite, connected graph. An arithmetical structure on G is a pair of positive integer vectors d, r such that (diag (d) - A) r=0 , where A is the adjacency matrix of G. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices (diag (d) - A)). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients ((2n-1)/(n-1)) , and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles
Counting Arithmetical Structures on Paths and Cycles
Let G be a finite, connected graph. An arithmetical structure on G is a pair of positive integer vectors d, r such that (diag (d) - A) r=0 , where A is the adjacency matrix of G. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices (diag (d) - A)). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients ((2n-1)/(n-1)) , and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles
Why Delannoy numbers?
This article is not a research paper, but a little note on the history of
combinatorics: We present here a tentative short biography of Henri Delannoy,
and a survey of his most notable works. This answers to the question raised in
the title, as these works are related to lattice paths enumeration, to the
so-called Delannoy numbers, and were the first general way to solve Ballot-like
problems. These numbers appear in probabilistic game theory, alignments of DNA
sequences, tiling problems, temporal representation models, analysis of
algorithms and combinatorial structures.Comment: Presented to the conference "Lattice Paths Combinatorics and Discrete
Distributions" (Athens, June 5-7, 2002) and to appear in the Journal of
Statistical Planning and Inference
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
Enumeration of Standard Young Tableaux
A survey paper, to appear as a chapter in a forthcoming Handbook on
Enumeration.Comment: 65 pages, small correction
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
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