Related papers: On Path diagrams and Stirling permutations
We investigate the number of permutations that occur in random labellings of trees. This is a generalisation of the number of subpermutations occurring in a random permutation. It also generalises some recent results on the number of…
In 1995, the first author introduced a multivariate generating function {$G$} that tracks the distribution of ascents and descents in labeled binary trees. In addition to proving that $G$ is symmetric, he conjectured that $G$ is Schur…
Stirling permutations were introduced by Gessel and Stanley, who used their enumeration by the number of descents to give a combinatorial interpretation of certain polynomials related to Stirling numbers. Quasi-Stirling permutations, which…
The transition matrix of a Markov chain $(X_k,k\geq 0)$ on a finite or infinite rooted tree is said to be almost upper-directed if, given $X_k$, the node $X_{k+1}$ is either a descendant of $X_k$ or the parent of $X_k$. It is said to be…
Recently proposed budding tree is a decision tree algorithm in which every node is part internal node and part leaf. This allows representing every decision tree in a continuous parameter space, and therefore a budding tree can be jointly…
We find a generating function expressed as a continued fraction that enumerates ordered trees by the number of vertices at different levels. Several Catalan problems are mapped to an ordered-tree problem and their generating functions also…
We study two related probabilistic models of permutations and trees biased by their number of descents. Here, a descent in a permutation $\sigma$ is a pair of consecutive elements $\sigma(i), \sigma(i+1)$ such that $\sigma(i) >…
We exploit a bijection between plane recursive trees and Stirling permutations; this yields the equivalence of some results previously proven separately by different methods for the two types of objects as well as some new results. We also…
Most well-known multidimensional continued fractions, including the M\"{o}nkemeyer map and the triangle map, are generated by repeatedly subdividing triangles. This paper constructs a family of multidimensional continued fractions by…
In this paper, we connect two types of representations of a permutation $\sigma$ of the finite field $\F_q$. One type is algebraic, in which the permutation is represented as the composition of degree-one polynomials and $k$ copies of…
We give an algorithm to explicitly compute the largest subtree, in the local Bruhat-Tits tree for PSL_2(k), whose vertices correspond to orders containing a given suborder H, in terms of a set of generators for H. The shape of this subtree…
We develop a new bijective framework for the enumeration of bipartite planar maps with control on the degree distribution of black and white vertices. Our approach builds on the blossoming-tree paradigm, introducing a family of orientations…
Andr\'e proved that the number of alternating permutations on $\{1, 2, \dots, n\}$ is equal to the Euler number $E_n$. A refinement of Andr\'e's result was given by Entringer, who proved that counting alternating permutations according to…
We study joint distributions of cycles and patterns in permutations written in standard cycle form. We explore both classical and generalised patterns of length 2 and 3. Many extensions of classical theory are achieved; bivariate generating…
The purpose of this article is to show how the isotropy subgroup of leaf permutations on binary trees can be used to systematically identify tree-informative invariants relevant to models of phylogenetic evolution. In the quartet case, we…
As a unification of increasing trees and plane trees, the weakly increasing trees labeled by a multiset was introduced by Lin-Ma-Ma-Zhou in 2021. Motived by some symmetries in plane trees proved recently by Dong, Du, Ji and Zhang, we…
In this paper, we count a dual set of Stirling permutations by the number of alternating runs. Properties of the generating functions, including recurrence relations, grammatical interpretations and convolution formulas are studied.
In this paper, we study the generating functions for the number of pattern restricted Stirling permutations with a given number of plateaus, descents and ascents. Properties of the generating functions, including symmetric properties and…
A permutation may be represented by a collection of paths in the plane. We consider a natural class of such representations, which we call tangles, in which the paths consist of straight segments at 45 degree angles, and the permutation is…
We describe a combinatorial approach for investigating properties of rational numbers. The overall approach rests on structural bijections between rational numbers and familiar combinatorial objects, namely rooted trees. We emphasize that…