# Stephanie van Willigenburg

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## Graduate Student Supervision

##### Doctoral Student Supervision

Dissertations completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest dissertations.

Extremal problems, in general, ask for the optimal size of certain finite objects when some restrictions are imposed. In extremal combinatorics, a major field in combinatorics, one studies how global properties guarantee the existence of local substructures, or equivalently, how avoiding local substructures poses a constraint on global quantities. In this thesis, we study the following three problems in combinatorics.The first problem is related to the Brown–Erdos–Sos conjecture, which can bereformulated as a problem of finding a dense substructure, namely a large numberof triples, spanned by a given number of elements in a quasigroup. In the special case of finite groups, we show in Chapter 2 that in every dense set of triples, there exists a subset of m elements which spans 4m/3 (1 − o(1)) triples, as m tends to infinity, which is much higher than the conjectured amount m − 3 for a general quasigroup. Later in the chapter, we give an elementary proof that, in finite groups, the maximum number of triples spanned by m elements has order m².The second problem concerns planar polygons in the 3-space. The maximum possible number of polygons from a given number of points is controlled by their intersection properties. Two hexagons in the space are said to intersect badly if their intersection consists of at least one common vertex as well as an interior point. In Chapter 3, we show that the number of hexagons on n points in 3-space without bad intersections is o(n²), under a mild assumption that the hexagons are ‘fat’. The main tool we used is the triangle removal lemma.The last problem in this dissertation is about the sum-product conjecture of Erdos and Szemeredi. The sum-product estimate concerns the larger size of the sumset and productset in terms of the size of the set itself. In Chapter 4 we prove an estimate with exponent 4/3 for the ring of quaternions and a certain family of well-conditioned matrices, using the boundedness of the kissing numbers.

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In this thesis, we make progress on the problem of enumerating tableaux on non-classicalshapes by introducing a general family of P-partitions that we call periodic P-partitions.Such a family of P-partitions generalizes the parallelogramic shapes, which were analysedby L´opez, Mart´ınez, P´erez, P´erez, Basova, Sun, Tewari, and van Willigenburg, and certaintruncated shifted shapes, where truncated shifted shapes were investigated by Adin, King,Roichman, and Panova. By introducing a separation property for posets and by proving arelationship between this property and P-partitions, we prove that periodic P-partitions canbe enumerated with a homogeneous first-order matrix difference equation.Afterwards, we consider families of finite sets that we call shellable and that have beencharacterized by Chang and by Hirst and Hughes as being the families of sets that admitunique solutions to Hall’s marriage problem. By introducing constructions on families of setsthat satisfy Hall’s Marriage Condition, and by using a combinatorial analogue of a shellingorder, we prove that shellable families can be characterized by using a generalized notionof hook-lengths. Then, we introduce a natural generalization of standard skew tableauxand Edelman and Greene’s balanced tableaux, then prove an existence result about such ageneralization using our characterization of shellable families.

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In this thesis, we study a natural noncommutative lift of the ubiquitous Schur functions, called noncommutative Schur functions. These functions were introduced by Bessenrodt, Luoto and van Willigenburg and resemble Schur functions in many regards. We prove some new results for noncommutative Schur functions that are analogues of classical results, and demonstrate that the resulting combinatorics in this setting is equally rich. First we prove a Murnaghan-Nakayama rule for noncommutative Schur functions. In other words, we give an explicit combinatorial formula for expanding the product of a noncommutative power sum symmetric function and a noncommutative Schur function in terms of noncommutative Schur functions. In direct analogy to the classical Murnaghan-Nakayama rule, the summands are computed using a noncommutative analogue of border strips, and have coefficients ±1 determined by the height of these border strips. The rule is proved by interpreting the noncommutative Pieri rules for noncommutative Schur functions in terms of box adding operators on compositions. We proceed to give a backward jeu de taquin slide analogue on semistandard reverse composition tableaux. These tableaux were first studied by Haglund, Luoto, Mason and van Willigenburg when defining quasisymmetric Schur functions. Our algorithm for performing backward jeu de taquin slides on semistandard reverse composition tableaux results in a natural operator on compositions that we call the jdt operator. This operator in turn gives rise to a new poset structure on compositions whose maximal chains we enumerate. As an application, we also give new right Pieri rules for noncommutative Schur functions that use the jdt operators, in contrast to the left Pieri rules given by Bessenrodt, Luoto and van Willigenburg.

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The Schur functions {s_lambda} and ubiquitous Littlewood-Richardson coefficients are instrumental in describing representation theory, symmetric functions,and even certain areas of algebraic geometry. Determining when two skewdiagrams D₁, D₂ have the same skew Schur function or determining when the difference of two such skew Schur functions SD₁ - SD₂ is Schur-positivereveals information about the structures corresponding to these functions.By defining a set of staircase diagrams that we can augment with other (skew) diagrams, we discover collections of skew diagrams for which the question of Schur-positivity among each difference can be resolved. Furthermore, for certain Schur-positive differences we give explicit formulas for computing the coefficients of the Schur functions in the difference.We extend from simple staircases to fat staircases, and carry on to diagrams called sums of fat staircases. These sums of fat staircases can also be augmented with other (skew) diagrams to obtain many instances of Schur positivity.We note that several of our Schur-positive differences become equalities of skew Schur functions when the number of variables is reduced. Finally, we give a factoring identity which allows one to obtain many of the non-trivial finite-variable equalities of skew Schur functions.

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##### Master's Student Supervision

Theses completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest theses.

A graph is Schur-positive if its chromatic symmetric function expands nonnegativelyin the Schur basis. All claw-free graphs are conjectured to be Schur-positive. Weintroduce a combinatorial object corresponding to a graph G, called a special rim hookG-tabloid, which is a variation on the special rim hook tabloid. These objects can beemployed to compute any Schur coefficient of the chromatic symmetric function of agraph. Special rim hook tabloids have previously been used to prove the non-Schur-positivity of some graphs. We construct sign-reversing maps on these special rimhook G-tabloids to prove that a family of claw-free graphs called generalized nets areSchur-positive. Thus, we demonstrate a new method for proving the Schur-positivityof graphs, which has the potential to be applied to make further progress toward theaforementioned conjecture.

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We consider the problem of when the difference of two ribbon Schur functions is a single Schur function. We prove that this near-equality phenomenon occurs in fourteen infinite families and we conjecture that these are the only possible cases. Towards this converse, we prove that under certain additional assumptions the only instances of near-equality are among our fourteen families. In particular, we prove that our first ten families are a complete classification of all cases where the difference of two ribbon Schur functions is a single Schur function whose corresponding partition has at most two parts at least 2. We also provide a framework for interpreting the remaining four families and we explore some ideas towards resolving our conjecture in general. We also determine some necessary conditions for the difference of two ribbon Schur functions to be Schur-positive.

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This work studies combinatorics related to expansions of a quasisymmetric refinementof Schur functions into Gessel’s fundamental basis, with almost all resultsconcerning whether an expansion is multiplicity-free. The combinatorial sideof this problem concerns a certain composition poset, and whether there are twostandard fillings of the same composition diagram with a given descent set. Thisthesis uses entirely combinatorial arguments, extending results by Bessenrodt andvanWilligenburg to work towards a classification of such descent multiplicity-freecompositions.The main tools used regard the situation of appending or prepending parts tocompositions. Compositions with multiplicity retain multiplicity when parts areappended or prepended, while multiplicity-free compositions stay multiplicity-freewhen a class of shapes called staircase-like are appended. A classification of compositionswhich are partitions or reverse partitions is achieved, leading up to aclassification of compositions not containing a part of length one. This is used asthe basis for a conjectured classification of multiplicity-free compositions withouta trailing staircase. The conjecture would in turn imply a complete classificationof multiplicity-free compositions.

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A major open problem in algebraic combinatorics is to find a combinatorial rule to compute the Kronecker product of two Schur functions. This is the same as decomposing the inner tensor product of two irreducible characters of the symmetric group as a sum of irreducible characters. Given that there is a combinatorial rule, namely the Littlewood-Richardson rule, which describes a way to compute the outer tensor product of two irreducible characters of the symmetric group, one would expect an algorithm which achieves the same purpose in the case of the inner tensor product. Jeffrey Remmel and Tamsen Whitehead first came up with a description of the Kronecker coefficients occurring in the Kronecker product of two Schur functions, both indexed by partitions of length at most 2. Mercedes Rosas later arrived at the same result using a different approach. The solution of the general problem would have implications in Complexity Theory and Quantum Information Theory.Our goal in this thesis is to derive formulae for computing the Kronecker product in certain cases where the Schur functions are indexed by partitions which are nearly rectangular. In particular, we study s{(n,n-1,1)}*s{(n,n)}, s{(n-1,n-1,1)}*s{(n,n-1)}, s{(n-1,n-1,2)}*s{(n,n)}, s{(n-1,n-1,1,1)}*s{(n,n)} and s{(n,n,1)}*s{(n,n,1)}. Our approach relies mainly on the fruitful interplay between manipulation of symmetric functions and the representation theory of the symmetric group. As a consequence of these formulae, we also derive an expression enumerating certain standard Young tableaux of bounded height.

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