11 papers
How to Use Deep Learning to Identify Sufficient Conditions: A Case Study on Stanley's -Positivity
Farid Aliniaeifard, Shu Xiao Li
In a study, published in Nature, researchers from DeepMind and mathematicians demonstrated a general framework using machine learning to make conjectures in pure mathematics. Here,…
The peak algebra in noncommuting variables
Farid Aliniaeifard, Shu Xiao Li
The well-known descent-to-peak map for the Hopf algebra of quasisymmetric functions, , and the peak algebra were originally defined by Stembr…
The extra basis in noncommuting variables
Farid Aliniaeifard, Stephanie van Willigenburg
We answer a question of Bergeron, Hohlweg, Rosas, and Zabrocki from 2006 to give a combinatorial description for the coproduct of the x-basis in the Hopf algebra of symmetric funct…
Hopf structures in the representation theory of direct products
Farid Aliniaeifard, Nathaniel Thiem
Combinatorial Hopf algebras give a linear algebraic structure to infinite families of combinatorial objects, a technique further enriched by the categorification of these structure…
Deletion-contraction for a unified Laplacian and applications
Farid Aliniaeifard, Victor Wang, Stephanie van Willigenburg
We define a graph Laplacian with vertex weights in addition to the more classical edge weights, which unifies the combinatorial Laplacian and the normalised Laplacian. Moreover, we…
Extended chromatic symmetric functions and equality of ribbon Schur functions
Farid Aliniaeifard, Victor Wang, Stephanie van Willigenburg
We prove a general inclusion-exclusion relation for the extended chromatic symmetric function of a weighted graph, which specializes to (extended) -deletion, and we give two met…