126 research outputs found
A closed formula for the number of convex permutominoes
In this paper we determine a closed formula for the number of convex
permutominoes of size n. We reach this goal by providing a recursive generation
of all convex permutominoes of size n+1 from the objects of size n, according
to the ECO method, and then translating this construction into a system of
functional equations satisfied by the generating function of convex
permutominoes. As a consequence we easily obtain also the enumeration of some
classes of convex polyominoes, including stack and directed convex
permutominoes
Solving multivariate functional equations
This paper presents a new method to solve functional equations of
multivariate generating functions, such as
giving a
formula for in terms of a sum over finite sequences. We use this
method to show how one would calculate the coefficients of the generating
function for parallelogram polyominoes, which is impractical using other
methods. We also apply this method to answer a question from fully commutative
affine permutations.Comment: 11 pages, 1 figure. v3: Main theorems and writing style revised for
greater clarity. Updated to final version, to appear in Discrete Mathematic
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
The perimeter generating functions of three-choice, imperfect, and 1-punctured staircase polygons
We consider the isotropic perimeter generating functions of three-choice,
imperfect, and 1-punctured staircase polygons, whose 8th order linear Fuchsian
ODEs are previously known. We derive simple relationships between the three
generating functions, and show that all three generating functions are joint
solutions of a common 12th order Fuchsian linear ODE. We find that the 8th
order differential operators can each be rewritten as a direct sum of a direct
product, with operators no larger than 3rd order. We give closed-form
expressions for all the solutions of these operators in terms of
hypergeometric functions with rational and algebraic arguments. The solutions
of these linear differential operators can in fact be expressed in terms of two
modular forms, since these hypergeometric functions can be expressed
with two, rational or algebraic, pullbacks.Comment: 28 page
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