<p>There are distinct differences between classes of matroids that are closed under principal extensions and those that are not Finite-field-representable matroids are not closed under principal extensions and they exhibit attractive properties like well-quasi-ordering and decidable theories (at least for subclasses with bounded branch-width). Infinite-field-representable matroids, on the other hand, are closed under principal extensions and exhibit none of these behaviours. For example, the class of rank-3 real representable matroids is not well-quasi-ordered and has an undecidable theory. The class of matroids that are transversal and cotransversal is not closed under principal extensions or coprincipal coextentions, so we expect it to behave more like the class of finite-field-representable matroids. This thesis is invested in exploring properties in the aforementioned class. A major idea that has inspired the thesis is the investigation of well-quasi-ordered classes in the world of matroids that are transversal and cotransversal. We conjecture that any minor-closed class with bounded branch-width containing matroids that are transversal and cotransversal is well-quasi-ordered. In Chapter 8 of the thesis, we prove this is true for lattice-path matroids, a well-behaved class that falls in this intersection. The general class of lattice-path matroids is not well-quasi-ordered as it contains an infinite antichain of so-called ‘notch matroids’. The interesting phenomenon that we observe is that this is essentially the only antichain in this class, that is, any minor-closed family of lattice-path matroids that contains only finitely many notch matroids is well-quasi-ordered. This answers a question posed by Jim Geelen. Another question that drove the research was recognising fundamental transversal matroids, since these matroids are also cotransversal. We prove that this problem in general is in NP and conjecture that it is NP-complete. We later explore this question for the classes of lattice-path and bicircular matroids. We are successful in finding polynomial-time algorithms in both classes that identify fundamental transversal matroids. We end this part by investigating the intersection of bicircular and cobicircular matroids. We define a specific class - whirly-swirls - and conjecture that eventually any matroid in the above mentioned intersection belongs to this class.</p>
<p>There are distinct differences between classes of matroids that are closed under principal extensions and those that are not Finite-field-representable matroids are not closed under principal extensions and they exhibit attractive properties like well-quasi-ordering and decidable theories (at least for subclasses with bounded branch-width). Infinite-field-representable matroids, on the other hand, are closed under principal extensions and exhibit none of these behaviours. For example, the class of rank-3 real representable matroids is not well-quasi-ordered and has an undecidable theory. The class of matroids that are transversal and cotransversal is not closed under principal extensions or coprincipal coextentions, so we expect it to behave more like the class of finite-field-representable matroids. This thesis is invested in exploring properties in the aforementioned class. A major idea that has inspired the thesis is the investigation of well-quasi-ordered classes in the world of matroids that are transversal and cotransversal. We conjecture that any minor-closed class with bounded branch-width containing matroids that are transversal and cotransversal is well-quasi-ordered. In Chapter 8 of the thesis, we prove this is true for lattice-path matroids, a well-behaved class that falls in this intersection. The general class of lattice-path matroids is not well-quasi-ordered as it contains an infinite antichain of so-called ‘notch matroids’. The interesting phenomenon that we observe is that this is essentially the only antichain in this class, that is, any minor-closed family of lattice-path matroids that contains only finitely many notch matroids is well-quasi-ordered. This answers a question posed by Jim Geelen. Another question that drove the research was recognising fundamental transversal matroids, since these matroids are also cotransversal. We prove that this problem in general is in NP and conjecture that it is NP-complete. We later explore this question for the classes of lattice-path and bicircular matroids. We are successful in finding polynomial-time algorithms in both classes that identify fundamental transversal matroids. We end this part by investigating the intersection of bicircular and cobicircular matroids. We define a specific class - whirly-swirls - and conjecture that eventually any matroid in the above mentioned intersection belongs to this class.</p>
Abstract
The lattice path model suggested by E. Deutsch is derived from ordinary Dyck paths, but with additional down-steps of size −3, −5, −7, . . . . For such paths, we find the generating functions of them, according to length, ending at level i, both, when considering them from left to right and from right to left. The generating functions are intrinsically cubic, and thus (for i = 0) in bijection to various objects, like even trees, ternary trees, etc.
Abstract
This paper considers sports matches in which two teams compete to score more points within a set amount of time (e.g. football, ice hockey). We focus on the order in which the competing teams score during the match (order of scoring). This type of order of scoring problem has not been addressed previously, and doing so here gives new insights into sports matches. For example, our analysis can deal with a situation that spectators find matches that involve comebacks particularly exciting. To describe such problems mathematically, we formulate the probabilities of (i) the favourite team leading throughout the match and (ii) the favourite team falling behind the opposing team but then making a comeback. These probabilities are derived using an independent Poisson model and lattice path enumeration, the latter of which involves the well-known ballot theorem. The independence assumption allows lattice path enumeration to be applied directly to the Poisson model and various scoring patterns to be addressed. We confirm that the values obtained from the proposed models agree well with actual sports data from football, futsal and ice hockey.
Following the work of Cano and Díaz, we consider a continuous analog of lattice path enumeration. This process allows us to define a continuous version of many discrete objects that count certain types of lattice paths. As an example of this process, we define continuous versions of binomial and multinomial coefficients, and describe some identities and partial differential equations that they satisfy. Finally, as an important byproduct of these continuous analogs, we illustrate a general method to recover discrete combinatorial quantities from their continuous analogs, via an application of the Khovanski-Puklikov discretizing Todd operators.
An identity for generating functions is proved in this paper. A novel method to compute the number of restricted lattice paths is developed on the basis of this identity. The method employs a difference equation with non-constant coefficients. Dyck paths, Schr¨oder paths, Motzkins path and other paths are computed to illustrate this method
On matroids that are transversal and cotransversal
