Computing the Residue Class of Partition Numbers

In 1919, Ramanujan initiated the study of congruence properties of the integer partition function $p(n)$ by showing that $$p(5n+4) \equiv 0 \mod{5}$$ and $$p(7n+5) \equiv 0 \mod{7}$$ hold for all integers $n$. These results attracted a lot of interest in the mathematical community and inspired other mathematicians to investigate the divisibility of various classes of integer partitions. The purpose of this thesis is to illustrate the use of generating series in the study of the residue classes of integer partition values. We begin by presenting the work of Mizuhara, Sellers and Swisher in 2015 on the residue classes of restricted plane partitions numbers. Next, we introduce Ramanujan's Conjecture regarding Ramanujan Congruences. Moreover, we use modular forms to present Ahlgren and Boylan's resolution of Ramanujan's Conjecture from 2003. Then, we discuss the open problems surrounding the distribution of the integer partitions values into residue classes and present Judge, Keith and Zanello's work from 2015 on the the distribution of the parity of the partition function. We continue by introducing $m-$ary partitions and provide an account of Andrews, Fraenkel and Sellers' work from 2015 and 2016 which yielded a complete characterization of the congruence classes of $m-$ary partitions with and without gaps. Finally, we present new results regarding the complete characterization of the residue classes of coloured $m-$ary partitions with and without gaps

Bernoulli measure on strings, and Thompson-Higman monoids

The Bernoulli measure on strings is used to define height functions for the dense R- and L-orders of the Thompson-Higman monoids M_{k,1}. The measure can also be used to characterize the D-relation of certain submonoids of M_{k,1}. The computational complexity of computing the Bernoulli measure of certain sets, and in particular, of computing the R- and L-height of an element of M_{k,1} is investigated.Comment: 27 pages

Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions

The m-Tamari lattice of F. Bergeron is an analogue of the clasical Tamari order defined on objects counted by Fuss-Catalan numbers, such as m-Dyck paths or (m+1)-ary trees. On another hand, the Tamari order is related to the product in the Loday-Ronco Hopf algebra of planar binary trees. We introduce new combinatorial Hopf algebras based on (m+1)-ary trees, whose structure is described by the m-Tamari lattices. In the same way as planar binary trees can be interpreted as sylvester classes of permutations, we obtain (m+1)-ary trees as sylvester classes of what we call m-permutations. These objects are no longer in bijection with decreasing (m+1)-ary trees, and a finer congruence, called metasylvester, allows us to build Hopf algebras based on these decreasing trees. At the opposite, a coarser congruence, called hyposylvester, leads to Hopf algebras of graded dimensions (m+1)^{n-1}, generalizing noncommutative symmetric functions and quasi-symmetric functions in a natural way. Finally, the algebras of packed words and parking functions also admit such m-analogues, and we present their subalgebras and quotients induced by the various congruences.Comment: 51 page

On Noncommutative Generalisations of Boolean Algebras

Skew Boolean algebras (SBA) and Boolean-like algebras (nBA) are one-pointed and n-pointed noncommutative generalisation of Boolean algebras, respectively. We show that any nBA is a cluster of n isomorphic right-handed SBAs, axiomatised here as the variety of skew star algebras. The variety of skew star algebras is shown to be term equivalent to the variety of nBAs. We use SBAs in order to develop a general theory of multideals for nBAs. We also provide a representation theorem for right-handed SBAs in terms of nBAs of n-partitions

The Congruences of a Finite Lattice, A "Proof-by-Picture" Approach, third edition

The major topic of this book: Congruence lattices of finite lattices. It covers about 80 years of research and 250 papers.Comment: Contains Part I of the boo