1,762 research outputs found
Signed Mahonians
A classical result of MacMahon gives a simple product formula for the
generating function of major index over the symmetric group. A similar
factorial-type product formula for the generating function of major index
together with sign was given by Gessel and Simion. Several extensions are given
in this paper, including a recurrence formula, a specialization at roots of
unity and type analogues.Comment: 23 page
A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics
A ``hybrid method'', dedicated to asymptotic coefficient extraction in
combinatorial generating functions, is presented, which combines Darboux's
method and singularity analysis theory. This hybrid method applies to functions
that remain of moderate growth near the unit circle and satisfy suitable
smoothness assumptions--this, even in the case when the unit circle is a
natural boundary. A prime application is to coefficients of several types of
infinite product generating functions, for which full asymptotic expansions
(involving periodic fluctuations at higher orders) can be derived. Examples
relative to permutations, trees, and polynomials over finite fields are treated
in this way.Comment: 31 page
Building Abelian Functions with Generalised Baker-Hirota Operators
We present a new systematic method to construct Abelian functions on Jacobian
varieties of plane, algebraic curves. The main tool used is a symmetric
generalisation of the bilinear operator defined in the work of Baker and
Hirota. We give explicit formulae for the multiple applications of the
operators, use them to define infinite sequences of Abelian functions of a
prescribed pole structure and deduce the key properties of these functions. We
apply the theory on the two canonical curves of genus three, presenting new
explicit examples of vector space bases of Abelian functions. These reveal
previously unseen similarities between the theories of functions associated to
curves of the same genus
On sets defining few ordinary lines
Let P be a set of n points in the plane, not all on a line. We show that if n
is large then there are at least n/2 ordinary lines, that is to say lines
passing through exactly two points of P. This confirms, for large n, a
conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for
this problem, as well as all sets having fewer than n - C ordinary lines for
some absolute constant C. We also solve, for large n, the "orchard-planting
problem", which asks for the maximum number of lines through exactly 3 points
of P. Underlying these results is a structure theorem which states that if P
has at most Kn ordinary lines then all but O(K) points of P lie on a cubic
curve, if n is sufficiently large depending on K.Comment: 72 pages, 16 figures. Third version prepared to take account of
suggestions made in a detailed referee repor
Manufacturing a mathematical group: a study in heuristics
I examine the way a relevant conceptual novelty in mathematics, that is, the notion of group, has been constructed in order to show the kinds of heuristic reasoning that enabled its manufacturing. To this end, I examine salient aspects of the works of Lagrange, Cauchy, Galois and Cayley (Sect. 2). In more detail, I examine the seminal idea resulting from Lagrange’s heuristics and how Cauchy, Galois and Cayley develop it. This analysis shows us how new mathematical entities are generated, and also how what counts as a solution to a problem is shaped and changed. Finally, I argue that this case study shows us that we have to study inferential micro-structures (Sect. 3), that is, the ways similarities and regularities are sought, in order to understand how theoretical novelty is constructed and heuristic reasoning is put forwar
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