https://www.math.tamu.edu/~frank.sottile/research/pdf/Loday.pdf

Structure of the Loday–Ronco Hopf algebra of trees

In recent years, plethystic calculus has emerged as a powerful technical tool for studying symmetric polynomials. In particular, some striking recent advances in the theory of Macdonald polynomials have relied heavily on plethystic computations. The main purpose of this article is to give a detailed explanation of a method for finding combinatorial interpretations of many commonly occurring plethystic expressions, which utilizes expansions in terms of quasisymmetric functions. To aid newcomers to plethysm, we also provide a self-contained exposition of the fundamental computational rules underlying plethystic calculus. Although these rules are well-known, their proofs can be difficult to extract from the literature. Our treatment emphasizes concrete calculations and the central role played by evaluation homomorphisms arising from the universal mapping property for polynomial rings.

/pdf/a-computational-and-combinatorial-expose-of-plethystic-45jwp11t9n.pdf

A computational and combinatorial exposé of plethystic calculus

Multiple zeta values have been studied by a wide variety of methods. In this article we summarize some of the results about them that can be obtained by an algebraic approach. This involves “coding” the multiple zeta values by monomials in two noncommuting variables x and y. Multiple zeta values can then be thought of as defining a map ζ: ℌ0 → R from a graded rational vector space ℌ0 generated by the “admissible words” of the noncommutative polynomial algebra Q〈x,y〉. Now ℌ0 admits two (commutative) products making ζ a homomorphism-the shuffle product and the “harmonic” product. The latter makes ℌ0 a subalgebra of the algebra QSym of quasi-symmetric functions. We also discuss some results about multiple zeta values that can be stated in terms of derivations and cyclic derivations of Q〈x,y〉, and we define an action of QSym on Q〈x,y〉 that appears useful. Finally, we apply the algebraic approach to relations of finite partial sums of multiple zeta value series.

Algebraic Aspects of Multiple Zeta Values

Loday and Ronco defined an interesting Hopf algebra structure on the linear span of the set of planar binary trees. They showed that the inclusion of the Hopf algebra of non-commutative symmetric functions in the Malvenuto-Reutenauer Hopf algebra of permutations factors through their Hopf algebra of trees, and these maps correspond to natural maps from the weak order on the symmetric group to the Tamari order on planar binary trees to the boolean algebra. 
We further study the structure of this Hopf algebra of trees using a new basis for it. We describe the product, coproduct, and antipode in terms of this basis and use these results to elucidate its Hopf-algebraic structure. We also obtain a transparent proof of its isomorphism with the non-commutative Connes-Kreimer Hopf algebra of Foissy, and show that this algebra is related to non-commutative symmetric functions as the (commutative) Connes-Kreimer Hopf algebra is related to symmetric functions.

Structure of the Loday-Ronco Hopf algebra of trees

Among the many significant contributions that Fishburn made to social choice theory some have focused on what he has called "acyclic sets", i.e. the sets of linear orders where majority rule applies without the "Condorcet effect" (majority relation never has cycles). The search for large domains of this type is a fascinating topic. I review the works in this field and in particular consider a recent one that allows to show the connections between some of them that have been unrelated up to now.

https://halshs.archives-ouvertes.fr/halshs-00198635/document

Acyclic Domains of Linear Orders: A Survey

An introduction to the complex representations of the symmetric group and general linear Lie algebra by P. Hanlon The cyclotomic identity by N. Metropolis and G. C. Rota Arrangements of hyperplanes and differential forms by P. Orlik, L. Solomon, and H. Terao Double centralizing theorems for wreath product by A. Regev On the construction of the maximal weight vectors in the tensor algebra of $\mathrm{gl}_n(\mathbb{C})$ by P. Hanlon The q-Dyson conjecture, generalized exponents, and the internal product of Schur functions by R. P. Stanley Spherical designs and group representations by E. Bannai Algorithms for plethysm by Y. M. Chen, A. M. Garsia, and J. Remmel Combinatorial correspondences for Young tableaux, and maximal chains in the weak Bruhat order of $S_n$ by P. Edelman and C. Greene Constructions on rim hook tableaux by D. E. White Orderings of Coxeter groups by A. Bjorner Modularly complemented lattices and shellability by D. Stanton and M. Wachs Counting faces and chains in polytopes and posets by M. M. Bayer and L. J. Billera The combinatorics of $(k,l)$-hook Schur functions by J. B. Remmel Multipartite P-partitions and inner products of skew Schur functions by I. M. Gessel Problem session.

Curtis Greene

Papers

Combinatorics and Algebra