A leader of a tree T on [n] is a vertex which has no smaller descendants in T . Gessel and Seo showed that ∑ T ∈Tn u(# of leaders in T )c(degree of 1 in T ) = uPn−1(1, u, cu), which is a generalization of Cayley’s formula, where Tn is the set of trees on [n] and Pn(a, b, c)= c n−1 ∏ i=1 ( ia + (n− i)b+ c. Using a variation of the Prüfer code which is called a RP-code, we give a simple bijective...

denote the ring of integer-valued polynomials on D with respect to the subset E (for ease of notation, if E = D, then set Int(D,D) = Int(D)). Gilmer’s work in this area (with the assistance of various co-authors) was truly groundbreaking and led to numerous extensions and generalizations by authors such as J. L. Chabert, P. J. Cahen, D. McQuillan and A. Loper. In this paper, we will review Gilm...

Sub-circuit recognition (SR) is a problem of recognizing sub-circuits within a given circuit and is a fundamental component in simulation, verification and testing of computeraided design. The SR problem can be formulated as subgraph isomorphism problem. Performance of previous works is not scalable as the complexities of modern designs increase. In this paper we propose a novel Prüfer-encoding...

We prove that the filling order is quadratic for a large class of solvable groups and asymptotically quadratic for all Q-rank one lattices in semisimple groups of R-rank at least 3. As a byproduct of auxiliary results we give a shorter proof of the theorem on the nondistorsion of horospheres providing also an estimate of a nondistorsion