Chromatic symmetric functions and change of basis

We introduce $H$-chromatic symmetric functions, $X_{G}^{H}$, which use the $H$-coloring of a graph $G$ to define a generalization of Stanley's chromatic symmetric functions. We say two graphs $G_1$ and $G_2$ are $H$-chromatically equivalent if $X_{G_1}^{H} = X_{G_2}^{H}$, and use this idea to study uniqueness results for $H$-chromatic symmetric functions, with a particular emphasis on the case $H$ is a complete bipartite graph. We also show that several of the classical bases of the space of symmetric functions, i.e. the monomial symmetric functions, power sum symmetric functions, and elementary symmetric functions, can be realized as $H$-chromatic symmetric functions. Moreover, we show that if $G$ and $H$ are particular types of multipartite complete graphs we can derive a set of $H$-chromatic symmetric functions that are a basis for $\Lambda^n$. We end with some conjectures and open problems.

This dissertation is dedicated to the study of positivity phenomena for the coefficients of the chromatic symmetric function of a graph. This function was introduced by Stanley in 1995 as a generalization of the chromatic polynomial of a graph. Stanley considered the expansion of the chromatic symmetric function in terms of various bases of symmetric functions, and conjectured the positivity of its coefficients in the basis of the elementary symmetric functions in the case of the incomparability graphs of (3 + 1)-free posets. The conjecture has not yet been proven, but has been checked for small graphs, and proven for certain families of graphs. The strongest general result in this direction was obtained by Gasharov. He proved a weaker statement, Schur positivity of the incomparability graphs of (3 + 1)-free posets. The strongest result on the positivity of the coefficients in the basis of the elementary symmetric functions was obtained by Stanley, who proved the positivity of certain sums of these coefficients by linking them to acyclic orientations of the incomparability graph. In this thesis we give a new proof of Gasharov’s theorem, which presents a combinatorial interpretation of the Schur-coefficients in terms of planar networks. Compared to Gasharov’s proof, it gives a clearer visual illustration of the cancellation procedures and is quite similar in spirit to the proof of monomial positivity of Schur functions via the Lindstrom–Gessel–Viennot Lemma. This construction led us to reconsider another idea of Stanley: instead of working with the chromatic symmetric function of a graph directly, we analyze certain analogs of the symmetric functions attached to graphs. We introduce a new combinatorial object: the correct sequences of unit interval orders, and using these, in certain cases, we succeed to construct combinatorial models of the coefficients appearing in Stanley’s conjecture. Our main result is the proof of positivity of the coefficients c_{n−k,1^k} , c_{n−2,2}, c_{n−3,2,1} and c_{2^k,1^{n−2k}} of the expansion of the chromatic symmetric function in terms of the basis of the elementary symmetric polynomials for the case of (3 + 1)-free posets.

We give a proof of the Stanley-Stembridge conjecture on chromatic symmetric functions for the class of all unit interval graphs with independence number 3. That is, we show that the chromatic symmetric function of the incomparability graph of a unit interval order in which the length of a chain is at most 3 is positively expanded as a linear sum of elementary symmetric functions.

On distinguishing trees by their chromatic symmetric functions

In Stanley’s seminal 1995 paper on the chromatic symmetric function, he stated that there was no known graph that was not contractible to the claw and whose chromatic symmetric function was not e -positive, that is, not a positive linear combination of elementary symmetric functions. We resolve this by giving infinite families of graphs that are not contractible to the claw and whose chromatic symmetric functions are not e -positive. Moreover, one such family is additionally claw-free, thus establishing that the e -positivity of chromatic symmetric functions is in general not dependent on the existence of an induced claw or of a contraction to a claw.

Chromatic symmetric functions of Dyck paths and [formula omitted]-rook theory

Let $h_\lambda$, $e_\lambda$, and $m_\lambda$ denote the homogeneous symmetric function, the elementary symmetric function and the monomial symmetric function associated with the partition $\lambda$ respectively. We give combinatorial interpretations for the coefficients that arise in expanding $m_\lambda$ in terms of homogeneous symmetric functions and the elementary symmetric functions. Such coefficients are interpreted in terms of certain classes of bi-brick permutations. The theory of Lyndon words is shown to play an important role in our interpretations.

The Jacobi-Stirling numbers and the Legendre-Stirling numbers of the first and second kind were first introduced by Everitt et al. (2002) and (2007) in the spectral theory. In this paper we note that Jacobi-Stirling numbers and Legendre-Stirling numbers are specializations of elementary and complete symmetric functions. We then study combinatorial interpretations of this specialization and obtain new combinatorial interpretations of the Jacobi-Stirling and Legendre-Stirling numbers.

Graphs with equal chromatic symmetric functions

Chromatic symmetric functions in noncommuting variables revisited

The complete and elementary symmetric functions are special cases of Schur functions. It is well-known that the Schur functions can be expressed in terms of complete or elementary symmetric functions using two determinant formulas: Jacobi–Trudi identity and Nagelsbach–Kostka identity. In this paper, we study new connections between complete and elementary symmetric functions.

MacMahon Symmetric Functions, the Partition Lattice, and Young Subgroups

We investigate an apparent hodgepodge of topics: a Robinson-Schensted algorithm for (3 + 1)-free posets, Chung and Graham's G-descent expansion of the chromatic polynomial, a quasi-symmetric expansion of the path-cycle symmetric function, and an expansion of Stanley's chromatic symmetric function X G in terms of a new symmetric function basis. We show how the theory of P-partitions (in particular, Stanley's quasi-symmetric function expansion of the chromatic symmetric function X G ) unifies them all, subsuming two old results and implying two new ones. Perhaps our most interesting result relates to the still-open problem of finding a Robinson-Schensted algorithm for (3 + 1)-free posets. (Magid has announced a solution but it appears to be incorrect.) We show that such an algorithm ought to “respect descents”, and that the best partial algorithm so far—due to Sundquist, Wagner, and West—respects descents if it avoids a certain induced subposet.

We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, and simplicial complexes. This coloring also generalizes oriented coloring, acyclic coloring, and star coloring. There is an associated symmetric function in noncommuting variables for which we give a deletion-contraction formula. In the case of graphs this symmetric function in noncommuting variables agrees with the chromatic symmetric function in noncommuting variables of Gebhard and Sagan. Our vertex coloring is a special case of the scheduling problems defined by Breuer and Klivans. We show how the deletion-contraction law can be applied to scheduling problems. Also, we show that the chromatic symmetric function determines the degree sequence of uniform hypertrees, but there exists pairs of 3-uniform hypertrees which are not isomorphic yet have the same chromatic symmetric function.

A generalization for the symmetry between complete symmetric functions and elementary symmetric functions is given. As corollaries we derive the inverse of a triangular Toeplitz matrix and the expression of the Toeplitz-Hessenberg determinant. A very large variety of identities involving integer partitions and multinomial coefficients can be generated using this generalization. The partitioned binomial theorem and a new formula for the partition function p(n) are obtained in this way.