Cеминары "Комбинаторика инвариантов Васильева": С.К.Ландо

Binomiality property and comultiplication

The binomial property expresses the polynomial $(x+y)^n$ as a sum of products of polynomials depending on $x$ and $y$ separately, the summation carried over partitions of~$n$ into a sum of two parts. There are many sequences of polynomials other than $x^n$ admitting a similar expression. In addition, many invariants possess a similar property, with the replacement of splitting of~$n$ into two parts by splitting of the combinatorial object in question into two subobjects. Such a property always suggests the existence of comultiplication and Hopf algebra structure on the vector space spanned by combinatorial objects under study.