MSRIUP 2013: Algebraic Combinatorics
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Number of permutations with same peak set for signed permutations
Team: Francis Castro, Jose Pastrana, Rita Zevallos
Supervised by: Alexander Diaz and Rosa Orellana
Let \( B_n \) be the group of all signed permutations of \( [n] \). A signed permutation has a peak in a position \(i=2, \ldots, n1\) if \( \pi_{i1} < \pi_i > \pi_{i+1} \). Let \( P(\pi) \) be the set of peaks of \( \pi \), \( P(S,n) \) be the set of signed permutations \( \pi \in B_n \) such that \( P(\pi)=S \), and \( \#P(S,n) \) be the cardinality of \( P(S,n) \). We show \( \#P(\emptyset,n)=2^{2n1} \) and \( \#P(S,n)=p(n)2^{2nS1} \) where \( p(n) \) is some polynomial. We also consider the case in which we add a zero at the beginning of the permutation to also allow peaks at position \( i=1 \).

The lattice of set partitions and transition matrices of symmetric functions
Team: Alexandria Burnley, Aquia Richburg, Simone Thiry
Supervised by: Rosa Orellana and Candice Price

Permutation Patterns for RealValued Functions
Team: Alicia Arrua, Gustavo MelendezRios, Lynesia Taylor
Supervised by: Samuel Ivy and Rosa Orellana

Consider the sequence \( [x, f(x), f(f(x)) = f^2(x), \ldots, f^{n1}(x)] \) where \( f \) is a realvalued function and \( n \geq 2\). We can associate a permutation to every such sequence by comparing it with \( x_1 < x_2 < ... < x_n \), where \( x_i = f^{j1}(x)\) for some \( j = 1, 2, \ldots , n\). Permutations that arise from these sequences are called allowed permutations and those that do not are called forbidden permutations.For example, the logistic map, \( f : [0,1] \rightarrow [0, 1] \) is defined by \(f(x) = rx(1 x)\) where \(0 \leq r \leq 4\), for any \(x\). We focus on enumerating the number of forbidden permutations for the logistic map and other functions, including trigonometric functions. For example, for the \(n = 3\) case, we have found that the oneline permutation (321) is a forbidden permutation for the function \( \sin(\pi x)\).

The Algebra of Set Partitions
Team: Ryan Contreras, Isabel Corona, Matt Sarmiento
Supervised by: Alexander Diaz and Rosa Orellana
A set partition of [n] = {1, 2, ...,n} is a collection of nonempty disjoint subsets of [n], called blocks, whose union is [n]. A block permutation of [n] consists of two set partitions A and B of [n] having the same number of blocks,and a bijection f : A > B. We consider the set BPn = {f : A > B f is a block permutation}. The elements in BPn can be visualized as graphs having two rows of n labeled vertices, corresponding to A and B. The connected components of each row are determined by connecting the vertices within each block of A and B. We then connect each block of A to the block of B which it maps to under f. The product g ยท f of two block permutations f : A > B and g : C > D of [n] is obtained by gluing the bottom of a graph representing f to the top of a graph representing g, and connecting each block of A to a block in D. We show that BPn is closed under this operation, and hence is a monoid. We have found a set of generators and seek to find a presentation for BPn. We also describe a Hopf algebra structure on BPn.

Chromatic Symmetric Functions of Trees and Unicycles
Team: Damien Gonzales, Arman Green, Caprice Stanley
Supervised by: Rosa Orellana and Candice Price
Given any simple graph, there is a corresponding symmetric function called the chromatic symmetric function (CSF). Introduced by Richard Stanley in 1995, the CSF of a graph G = (V(G), E(G)) is defined as follows: \[ X_G = \sum_{\kappa} \prod_{v \in V(G)} x_{\kappa (v)}\] where the sum is over all proper colorings \( \kappa \) of G. A proper coloring is a labeling of a graph such that no two adjacent vertices have the same label. In Geoffrey Scott's senior thesis, published in 2008, several open problems in graph theory were presented. In our poster, we investigate these open problems and generalize some of his results. Our goal is to find necessary conditions for any two graphs that will ensure that they have the same CSF. We first consider two special types of graphs: trees and unicycles and write a program in SAGE to compare the CSF for any simple graph, and thus, compile a library of graphs with a small number of vertices alongside their CSFs.

On the Schur Positivity of Differences of Products of Schur Functions
Team: Jeremiah Emidih, Nadine Jensen, Jeremy Meza
Supervised by: Samuel Ivy and Rosa Orellana
The Schur functions are a basis for the ring of symmetric functions indexed by partitions of nonnegative integers. A symmetric function f is called Schur positive if when expressed as a linear combination of Schur functions $$f=\sum_{\lambda}c_{\lambda}s_{\lambda}$$ each coefficient \( c_{\lambda} \) is nonnegative. We wish to investigate expressions of the form $$s_{\lambda^c}s_{\lambda}s_{\mu^c}s_{\mu}$$ where \( \lambda \) partitions n and \( \mu \) partitions n1 and the complements \( \lambda^c, \mu^c \) are taken over a sufficiently large \( m \times m \) square. We give a necessary condition that if (1) is Schur positive, then \( \mu \) is contained in \( \lambda \). Furthermore, we show how conjugating partitions preserve Schur positivity. Lastly, we incorporate the Littlewood Richardson rule to show that particular classes of \( \lambda \) of \( \mu \) are never Schur positive.