Generalized Cauchy identities, trees and multidimensional Brownian motions. Part II: Combinatorial differential calculus

TL;DR

This paper develops a combinatorial differential calculus using ordered sets and trees, enabling analytic proofs of generalized Cauchy identities and providing a framework to encode additional information in combinatorial structures.

Contribution

It introduces a novel combinatorial calculus framework that parallels traditional calculus, allowing for analytic proofs and encoding extra data in combinatorial objects.

Findings

Reformulation of generalized Cauchy identities using combinatorial calculus

Unique determination of bijections via additional vertex information

Extension of calculus concepts to combinatorial structures

Abstract

We present an analogue of the differential calculus in which the role of polynomials is played by certain ordered sets and trees. Our combinatorial calculus has all nice features of the usual calculus and has an advantage that the elements of the considered ordered sets might carry some additional information. In this way an analytic proof of generalized Cauchy identities from the previous work of the second author can be directly reformulated in our new language of the combinatorial calculus; furthermore the additional information carried by vertices determines uniquely the bijections presented in Part I of this series.