Abstract
We describe the relation between graph decompositions into walks and the normal ordering of differential operators in the $n$-th Weyl algebra. Under several specifications, we study new types of restricted set partitions, and a generalization of Stirling numbers, which we call the $\lambda$-Stirling numbers.
Highlights
Let G = (V, E) be a digraph with an ordered set of edges E = (e1, . . . , em)
We describe the relation between graph decompositions into walks and the normal ordering of differential operators in the n-th Weyl algebra
We study new types of restricted set partitions, and a generalization of Stirling numbers, which we call the λ-Stirling numbers
Summary
If G is a path and edges along the path are labeled by an arbitrary permutation σ ∈ Sm, decompositions into minimal number of increasing walks index the descent set of σ This interpretation arises from the normal ordering problem in the Weyl algebra. This corresponds to the normal ordering x1∂2x2∂3x3∂1 = x1∂1 + x1x3∂3∂1 + x1x2∂2∂1 + x1x2x3∂2∂3∂1, where the sources and sinks of walks are exactly the indices of terms. In this paper we focus on combinatorial aspects of walk decompositions and various specifications of the normal ordering interpretations, such as restricted set partitions and generalized Stirling numbers. We obtain many properties of Sλ(n, k), Cλ(n, k) analogous to the properties of the usual Stirling numbers
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
