1,542 research outputs found
What power of two divides a weighted Catalan number?
Given a sequence of integers b = (b_0,b_1,b_2,...) one gives a Dyck path P of
length 2n the weight
wt(P) = b_{h_1} b_{h_2} ... b_{h_n},
where h_i is the height of the ith ascent of P. The corresponding weighted
Catalan number is
C_n^b = sum_P wt(P),
where the sum is over all Dyck paths of length 2n. So, in particular, the
ordinary Catalan numbers C_n correspond to b_i = 1 for all i >= 0. Let xi(n)
stand for the base two exponent of n, i.e., the largest power of 2 dividing n.
We give a condition on b which implies that
xi(C_n^b) = xi(C_n).
In the special case b_i=(2i+1)^2, this settles a conjecture of Postnikov
about the number of plane Morse links. Our proof generalizes the recent
combinatorial proof of Deutsch and Sagan of the classical formula for xi(C_n).Comment: Fixed reference
A combinatorial approach to the power of 2 in the number of involutions
We provide a combinatorial approach to the largest power of in the number
of permutations with , for a fixed prime number . With this
approach, we find the largest power of in the number of involutions, in the
signed sum of involutions and in the numbers of even or odd involutions.Comment: 13 page
Divisors and specializations of Lucas polynomials
Three-term recurrences have infused stupendous amount of research in a broad
spectrum of the sciences, such as orthogonal polynomials (in special functions)
and lattice paths (in enumerative combinatorics). Among these are the Lucas
polynomials, which have seen a recent true revival. In this paper one of the
themes of investigation is the specialization to the Pell and Delannoy numbers.
The underpinning motivation comprises primarily of divisibility and symmetry.
One of the most remarkable findings is a structural decomposition of the Lucas
polynomials into what we term as flat and sharp analogs.Comment: Minor typos are fixed, new references are added. To appear in Journal
of Combinatoric
Donaldson-Thomas invariants, torus knots, and lattice paths
In this paper we find and explore the correspondence between quivers, torus
knots, and combinatorics of counting paths. Our first result pertains to quiver
representation theory -- we find explicit formulae for classical generating
functions and Donaldson-Thomas invariants of an arbitrary symmetric quiver. We
then focus on quivers corresponding to torus knots and show that their
classical generating functions, in the extremal limit and framing , are
generating functions of lattice paths under the line of the slope .
Generating functions of such paths satisfy extremal A-polynomial equations,
which immediately follows after representing them in terms of the Duchon
grammar. Moreover, these extremal A-polynomial equations encode
Donaldson-Thomas invariants, which provides an interesting example of
algebraicity of generating functions of these invariants. We also find a
quantum generalization of these statements, i.e. a relation between motivic
quiver generating functions, quantum extremal knot invariants, and -weighted
path counting. Finally, in the case of the unknot, we generalize this
correspondence to the full HOMFLY-PT invariants and counting of Schr\"oder
paths.Comment: 45 pages. Corrected typos in new versio
Determinants of (generalised) Catalan numbers
We show that recent determinant evaluations involving Catalan numbers and
generalisations thereof have most convenient explanations by combining the
Lindstr\"om-Gessel-Viennot theorem on non-intersecting lattice paths with a
simple determinant lemma from [Manuscripta Math. 69 (1990), 173-202]. This
approach leads also naturally to extensions and generalisations.Comment: AmS-TeX, 16 pages; minor correction
Lower order terms in the 1-level density for families of holomorphic cuspidal newforms
The Katz-Sarnak density conjecture states that, in the limit as the
conductors tend to infinity, the behavior of normalized zeros near the central
point of families of L-functions agree with the N -> oo scaling limits of
eigenvalues near 1 of subgroups of U(N). Evidence for this has been found for
many families by studying the n-level densities; for suitably restricted test
functions the main terms agree with random matrix theory. In particular, all
one-parameter families of elliptic curves with rank r over Q(T) and the same
distribution of signs of functional equations have the same limiting behavior.
We break this universality and find family dependent lower order correction
terms in many cases; these lower order terms have applications ranging from
excess rank to modeling the behavior of zeros near the central point, and
depend on the arithmetic of the family. We derive an alternate form of the
explicit formula for GL(2) L-functions which simplifies comparisons, replacing
sums over powers of Satake parameters by sums of the moments of the Fourier
coefficients lambda_f(p). Our formula highlights the differences that we expect
to exist from families whose Fourier coefficients obey different laws (for
example, we expect Sato-Tate to hold only for non-CM families of elliptic
curves). Further, by the work of Rosen and Silverman we expect lower order
biases to the Fourier coefficients in families of elliptic curves with rank
over Q(T); these biases can be seen in our expansions. We analyze several
families of elliptic curves and see different lower order corrections,
depending on whether or not the family has complex multiplication, a forced
torsion point, or non-zero rank over Q(T).Comment: 38 pages, version 2.2: fixed some typos, included some comments from
Steven Finch which give more rapidly converging expressions for the constants
gamma_{PNT}, gamma_{PNT,1,3} and gamma_{PNT,1,4}, updated reference
- …