A tableau inversion is a pair of entries in row-standard tableau T that lie in the same column of... more A tableau inversion is a pair of entries in row-standard tableau T that lie in the same column of T yet lack the appropriate relative ordering to make T column-standard. An i-inverted Young tableau is a row-standard tableau along with a precisely i inversion pairs. Tableau inversions were originally introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, with the number of i-inverted tableaux that standardize to a fixed standard Young tableau corresponding to a specific Betti number of the associated fiber. In this paper we approach the topic of tableau inversions from a completely combinatorial perspective. We develop formulas enumerating the number of i-inverted Young tableaux for a variety of tableaux shapes, not restricting ourselves to inverted tableaux that standardize a specific standard Young tableau, and construct bijections between i-inverted Young tableaux of a certain shape with j-inverted Young tableaux of different shapes. Finally, we share so...
Deliverables for IPRO 301: Undergraduate Research on Interprofessional Education for the Fall 200... more Deliverables for IPRO 301: Undergraduate Research on Interprofessional Education for the Fall 2007 semester
In 1935, Paul Erdős and George Szekeres were able to show that any point set large enough contain... more In 1935, Paul Erdős and George Szekeres were able to show that any point set large enough contains the vertices of a convex k-gon. Later in 1961, they constructed a point set of size 2k−2 not containing the vertex set of any convex k-gon. This leads to what is known as the Erdős-Szekeres Conjecture, that any point set of 2k−2 + 1 points contains the vertices of a convex k-gon. Recently, this famous problem of planar geometry has been transformed into a problem of finding cliques in a graph of copoints. We will discuss results and open problems corresponding to this graph of copoints.
The Raney numbers Rp,r(k) are a two-parameter generalization of the Catalan numbers that were int... more The Raney numbers Rp,r(k) are a two-parameter generalization of the Catalan numbers that were introduced by Raney in his investigation of functional composition patterns. We give a new combinatorial interpretation for the Raney numbers in terms of planar embeddings of certain collections of trees, a construction that recovers the usual interpretation of the p-Catalan numbers in terms of p-ary trees via the specialization Rp,1(k) = pck. Our technique leads to several combinatorial identities involving the Raney numbers and ordered partitions. We then give additional combinatorial interpretations of specific Raney numbers, including an identification of Rp2,p(k) with oriented trees whose vertices satisfy the “source-sink property.” We close with comments applying these results to the enumeration of connected (non-elliptic) A2 webs that lack an internal cycle.
Many colleges have regularly occurring math seminars or math club meetings that seek to deepen st... more Many colleges have regularly occurring math seminars or math club meetings that seek to deepen students’ knowledge and interest in mathematics through engaging activities. At Valparaiso University, our weekly mathematics colloquium typically concludes the fall semester with a holiday-themed activity. Popular past activities have included counting the total number of gifts given in the song “The Twelve Days of Christmas” and computing the area and perimeter of the Koch snowflake. With “The Twelve Days of Christmas” song, students can explore connections to Pascal’s triangle [5] and generalize the formula for the total number of gifts given if the number of days is not restricted to equal twelve [4]. With the Koch snowflake, students can explore properties of fractals and practice developing recursive formulas and computing limits of infinite sequences and series [6]. The advantage of these activities is that they are accessible to freshmen, while still holding the interest of senior mathematics majors. Because our majors enroll in mathematics colloquium all four years of their undergraduate studies, we sought to create additional holiday-themed math activities in order for upperclassmen to avoid repeating previous content. In this article, we define a new class of trees called “Christmas trees” and present material regarding Christmas trees that could be incorporated into an engaging class activity.
A tableau inversion is a pair of entries in row-standard tableau $T$ that lie in the same column ... more A tableau inversion is a pair of entries in row-standard tableau $T$ that lie in the same column of $T$ yet lack the appropriate relative ordering to make $T$ column-standard. An $i$-inverted Young tableau is a row-standard tableau along with precisely $i$ inversion pairs. Tableau inversions were originally introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, with the number of $i$-inverted tableaux that standardize to a fixed standard Young tableau corresponding to a specific Betti number of the associated fiber. In this paper we approach the topic of tableau inversions from a completely combinatorial perspective. We develop formulas enumerating the number of $i$-inverted Young tableaux for a variety of tableaux shapes, not restricting ourselves to inverted tableau that standardize a specific standard Young tableau, and construct bijections between $i$-inverted Young tableaux of a certain shape with $j$-inverted Young tableaux of different shapes. Fina...
Abstract Homework is prevalent in mathematics courses, as are cumulative final exams. This study ... more Abstract Homework is prevalent in mathematics courses, as are cumulative final exams. This study incorporated the memory science concepts of the testing effect and spacing effect in the homework and final exam of college mathematics courses. By replacing some new homework problems with review problems, students had additional opportunities to recall old material. Effects were analyzed by comparing the final exam scores of randomized groups of students, which showed small positive gains for students who had the experimental homework design. Additionally, students were split into the categories of low-scorers and high-scorers based on their first test score, prior to the intervention. Low-scoring students saw more benefits than the high-scoring students.
A tableaux inversion is a pair of entries in row-standard tableaux $T$ that lie in the same colum... more A tableaux inversion is a pair of entries in row-standard tableaux $T$ that lie in the same column of $T$ yet lack the appropriate relative ordering to make $T$ column-standard. An $i$-inverted Young tableaux is a row-standard tableaux along with a precisely $i$-inversion pairs. Tableaux inversions were originally introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, with the number of $i$-inverted tableaux that standardize to a fixed standard Young tableaux corresponding to a specific Betti number of the associated fiber. In this paper we approach the topic of tableaux inversions from a completely combinatorial perspective. We develop formulas enumerating the number of $i$-inverted Young tableaux for a variety of tableaux shapes, not restricting ourselves to inverted tableaux that standardize a specific standard Young tableaux, and construct bijections between $i$-inverted Young tableaux of a certain shape with $j$-inverted Young tableaux of different sh...
The Raney numbers $R_{p,r}(n)$ are a two-parameter generalization of the Catalan numbers that wer... more The Raney numbers $R_{p,r}(n)$ are a two-parameter generalization of the Catalan numbers that were introduced by Raney in his investigation of functional composition patterns \cite{Raney}. We give a new combinatorial interpretation for all Raney numbers in terms of planar embeddings of certain collections of trees, a construction that recovers the usual interpretation of the $p$-Catalan numbers in terms of $p$-ary trees via the specialization $R_{p,1}(n) =_{p} c_n$. Our technique leads to several combinatorial identities involving the Raney numbers and ordered partitions. We then give additional combinatorial interpretations of specific Raney numbers, including an identification of $R_{p^2,p}(n)$ with oriented trees whose vertices satisfy the "source or sink property". We close with comments applying these results to the enumeration of connected (non-elliptic) $A_2$ webs that lack an internal cycle.
Given a Hermitian matrix A 2 Mn(C), associate a simple, undirected graph G(A) where V (G) = {1,2,... more Given a Hermitian matrix A 2 Mn(C), associate a simple, undirected graph G(A) where V (G) = {1,2,...,n} and E(G) = { ij | aij 6= 0,i 6= j }. The collection of all Hermitian matrices that share a common graph G is denoted H(G). The problem of finding the multiplicities of the eigenvalues among the matrices in H(G) has
Introduction Chicago Public Schools (CPS) annually hold a science fair competition requiring all ... more Introduction Chicago Public Schools (CPS) annually hold a science fair competition requiring all high school students to create and present a science fair project. At the onset of the science fair process, there are local school-wide science fairs. The best student science fair ...
A tableau inversion is a pair of entries in row-standard tableau T that lie in the same column of... more A tableau inversion is a pair of entries in row-standard tableau T that lie in the same column of T yet lack the appropriate relative ordering to make T column-standard. An i-inverted Young tableau is a row-standard tableau along with a precisely i inversion pairs. Tableau inversions were originally introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, with the number of i-inverted tableaux that standardize to a fixed standard Young tableau corresponding to a specific Betti number of the associated fiber. In this paper we approach the topic of tableau inversions from a completely combinatorial perspective. We develop formulas enumerating the number of i-inverted Young tableaux for a variety of tableaux shapes, not restricting ourselves to inverted tableaux that standardize a specific standard Young tableau, and construct bijections between i-inverted Young tableaux of a certain shape with j-inverted Young tableaux of different shapes. Finally, we share so...
Deliverables for IPRO 301: Undergraduate Research on Interprofessional Education for the Fall 200... more Deliverables for IPRO 301: Undergraduate Research on Interprofessional Education for the Fall 2007 semester
In 1935, Paul Erdős and George Szekeres were able to show that any point set large enough contain... more In 1935, Paul Erdős and George Szekeres were able to show that any point set large enough contains the vertices of a convex k-gon. Later in 1961, they constructed a point set of size 2k−2 not containing the vertex set of any convex k-gon. This leads to what is known as the Erdős-Szekeres Conjecture, that any point set of 2k−2 + 1 points contains the vertices of a convex k-gon. Recently, this famous problem of planar geometry has been transformed into a problem of finding cliques in a graph of copoints. We will discuss results and open problems corresponding to this graph of copoints.
The Raney numbers Rp,r(k) are a two-parameter generalization of the Catalan numbers that were int... more The Raney numbers Rp,r(k) are a two-parameter generalization of the Catalan numbers that were introduced by Raney in his investigation of functional composition patterns. We give a new combinatorial interpretation for the Raney numbers in terms of planar embeddings of certain collections of trees, a construction that recovers the usual interpretation of the p-Catalan numbers in terms of p-ary trees via the specialization Rp,1(k) = pck. Our technique leads to several combinatorial identities involving the Raney numbers and ordered partitions. We then give additional combinatorial interpretations of specific Raney numbers, including an identification of Rp2,p(k) with oriented trees whose vertices satisfy the “source-sink property.” We close with comments applying these results to the enumeration of connected (non-elliptic) A2 webs that lack an internal cycle.
Many colleges have regularly occurring math seminars or math club meetings that seek to deepen st... more Many colleges have regularly occurring math seminars or math club meetings that seek to deepen students’ knowledge and interest in mathematics through engaging activities. At Valparaiso University, our weekly mathematics colloquium typically concludes the fall semester with a holiday-themed activity. Popular past activities have included counting the total number of gifts given in the song “The Twelve Days of Christmas” and computing the area and perimeter of the Koch snowflake. With “The Twelve Days of Christmas” song, students can explore connections to Pascal’s triangle [5] and generalize the formula for the total number of gifts given if the number of days is not restricted to equal twelve [4]. With the Koch snowflake, students can explore properties of fractals and practice developing recursive formulas and computing limits of infinite sequences and series [6]. The advantage of these activities is that they are accessible to freshmen, while still holding the interest of senior mathematics majors. Because our majors enroll in mathematics colloquium all four years of their undergraduate studies, we sought to create additional holiday-themed math activities in order for upperclassmen to avoid repeating previous content. In this article, we define a new class of trees called “Christmas trees” and present material regarding Christmas trees that could be incorporated into an engaging class activity.
A tableau inversion is a pair of entries in row-standard tableau $T$ that lie in the same column ... more A tableau inversion is a pair of entries in row-standard tableau $T$ that lie in the same column of $T$ yet lack the appropriate relative ordering to make $T$ column-standard. An $i$-inverted Young tableau is a row-standard tableau along with precisely $i$ inversion pairs. Tableau inversions were originally introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, with the number of $i$-inverted tableaux that standardize to a fixed standard Young tableau corresponding to a specific Betti number of the associated fiber. In this paper we approach the topic of tableau inversions from a completely combinatorial perspective. We develop formulas enumerating the number of $i$-inverted Young tableaux for a variety of tableaux shapes, not restricting ourselves to inverted tableau that standardize a specific standard Young tableau, and construct bijections between $i$-inverted Young tableaux of a certain shape with $j$-inverted Young tableaux of different shapes. Fina...
Abstract Homework is prevalent in mathematics courses, as are cumulative final exams. This study ... more Abstract Homework is prevalent in mathematics courses, as are cumulative final exams. This study incorporated the memory science concepts of the testing effect and spacing effect in the homework and final exam of college mathematics courses. By replacing some new homework problems with review problems, students had additional opportunities to recall old material. Effects were analyzed by comparing the final exam scores of randomized groups of students, which showed small positive gains for students who had the experimental homework design. Additionally, students were split into the categories of low-scorers and high-scorers based on their first test score, prior to the intervention. Low-scoring students saw more benefits than the high-scoring students.
A tableaux inversion is a pair of entries in row-standard tableaux $T$ that lie in the same colum... more A tableaux inversion is a pair of entries in row-standard tableaux $T$ that lie in the same column of $T$ yet lack the appropriate relative ordering to make $T$ column-standard. An $i$-inverted Young tableaux is a row-standard tableaux along with a precisely $i$-inversion pairs. Tableaux inversions were originally introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, with the number of $i$-inverted tableaux that standardize to a fixed standard Young tableaux corresponding to a specific Betti number of the associated fiber. In this paper we approach the topic of tableaux inversions from a completely combinatorial perspective. We develop formulas enumerating the number of $i$-inverted Young tableaux for a variety of tableaux shapes, not restricting ourselves to inverted tableaux that standardize a specific standard Young tableaux, and construct bijections between $i$-inverted Young tableaux of a certain shape with $j$-inverted Young tableaux of different sh...
The Raney numbers $R_{p,r}(n)$ are a two-parameter generalization of the Catalan numbers that wer... more The Raney numbers $R_{p,r}(n)$ are a two-parameter generalization of the Catalan numbers that were introduced by Raney in his investigation of functional composition patterns \cite{Raney}. We give a new combinatorial interpretation for all Raney numbers in terms of planar embeddings of certain collections of trees, a construction that recovers the usual interpretation of the $p$-Catalan numbers in terms of $p$-ary trees via the specialization $R_{p,1}(n) =_{p} c_n$. Our technique leads to several combinatorial identities involving the Raney numbers and ordered partitions. We then give additional combinatorial interpretations of specific Raney numbers, including an identification of $R_{p^2,p}(n)$ with oriented trees whose vertices satisfy the "source or sink property". We close with comments applying these results to the enumeration of connected (non-elliptic) $A_2$ webs that lack an internal cycle.
Given a Hermitian matrix A 2 Mn(C), associate a simple, undirected graph G(A) where V (G) = {1,2,... more Given a Hermitian matrix A 2 Mn(C), associate a simple, undirected graph G(A) where V (G) = {1,2,...,n} and E(G) = { ij | aij 6= 0,i 6= j }. The collection of all Hermitian matrices that share a common graph G is denoted H(G). The problem of finding the multiplicities of the eigenvalues among the matrices in H(G) has
Introduction Chicago Public Schools (CPS) annually hold a science fair competition requiring all ... more Introduction Chicago Public Schools (CPS) annually hold a science fair competition requiring all high school students to create and present a science fair project. At the onset of the science fair process, there are local school-wide science fairs. The best student science fair ...
Uploads
Papers by Jonathan Beagley