On 5-Regular Bipartitions with even Parts Distinct

In 2010, Andrews, Michael D. Hirschhorn and James A. Sellers considered the function ped(n), the number of partition of an integer n with even parts distinct (the odd parts are unrestricted). They obtained infinite families of congruences in the spirit of Ramanujan's congruences for the unrestricted partition function p(n). Let b(n) denote the number of 5-regular bipartitions of a positive integer n with even parts distinct (odd parts are unrestricted). In this paper, we establish many infinite families of congruences modulo powers of 2 for b(n). For example, ∑ n = 0 ∞ b 16 · 3 2 α · 5 2 β n + 14 · 3 2 α · 5 2 β + 1 q n = 8 f 2 3 f 5 3 ( mod 16 ) , where α , β ≥ 0

On 5-Regular Bipartitions with even Parts Distinct

In 2010, Andrews, Michael D. Hirschhorn and James A. Sellers considered the function ped(n), the number of partition of an integer n with even parts distinct (the odd parts are unrestricted). They obtained infinite families of congruences in the spirit of Ramanujan's congruences for the unrestricted partition function p(n). Let b(n) denote the number of 5-regular bipartitions of a positive integer n with even parts distinct (odd parts are unrestricted). In this paper, we establish many infinite families of congruences modulo powers of 2 for b(n). For example, ∑ n = 0 ∞ b 16 · 3 2 α · 5 2 β n + 14 · 3 2 α · 5 2 β + 1 q n = 8 f 2 3 f 5 3 ( mod 16 ) , where α , β ≥ 0

Congruences for (2, 3)-regular partition with designated summands

Let $PD_{2, 3}(n)$ count the number of partitions of $n$ with designated summands in which parts are not multiples of $2$ or $3$. In this work, we establish congruences modulo powers of 2 and 3 for $PD_{2, 3}(n)$. For example, for each \quad $n\ge0$ and $\alpha\geq0$ \quad $PD_{2, 3}(6\cdot4^{\alpha+2}n+5\cdot4^{\alpha+2})\equiv 0 \pmod{2^4}$ and $PD_{2, 3}(4\cdot3^{\alpha+3}n+10\cdot3^{\alpha+2})\equiv 0 \pmod{3}.

ARITHMETIC PROPERTIES OF PARTITIONS AND HECKE-ROGERS TYPE IDENTITIES

Ph.DDOCTOR OF PHILOSOPH

A result on the c2c_2 invariant for powers of primes

The $c_2$ invariant is an arithmetic graph invariant related to quantum field theory. We give a relation modulo $p$ between the $c_2$ invariant at $p$ and the $c_2$ invariant at $p^s$, providing evidence for a conjecture of Schnetz. The key result is a relation modulo $p$ between certain coefficients of powers of products of particularly nice polynomials.Comment: 16 pages, edits according to referee comments including extracting the main result at the level of polynomial

Coxeter systems, multiplicity free representations, and twisted Kazhdan-Lusztig Theory

Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (pages 195-201).This thesis considers three topics related to the representations of Coxeter systems, their Hecke algebras, and related groups. The first topic concerns the construction of generalized involution models, as defined by Bump and Ginzburg. We compute the automorphism groups of all complex reflection groups G(r, p, n) and using this information, we classify precisely which complex reflection groups have generalized involution models. The second topic concerns the set of "unipotent characters" Uch(W) which Lusztig has attached to each finite, irreducible Coxeter system (W, S). We describe a precise sense in which the irreducible multiplicities of a certain W-representation can be used to define a function which serves naturally as a heuristic definition of the Frobenius-Schur indicator on Uch(W). The formula we obtain for this indicator extends prior work of Casselman, Kottwitz, Lusztig, and Vogan addressing the case in which W is a Weyl group. Finally, we study a certain module of the Hecke algebra of a Coxeter system (W, S), spanned by the set of twisted involutions in W. Lusztig has shown that this module has two distinguished bases, and that the transition matrix between these bases defines interesting analogs of the much-studied Kazhdan-Lusztig polynomials of (W, S). We prove several positivity properties related to these polynomials for universal Coxeter systems, using combinatorial techniques, and for finite Coxeter systems, using computational methods.by Eric Marberg.Ph.D