Related papers: A Combinatorial Identity for Rooted Labeled Forest…
Random forests are decision tree ensembles that can be used to solve a variety of machine learning problems. However, as the number of trees and their individual size can be large, their decision making process is often incomprehensible. In…
We obtain a generating function for the degree sequences and colors of rooted multipartite labeled series-reduced trees. As an application of this result, we determine the number of symbolic ultrametrics (introduced by B\"ocker and Dress)…
In a recent paper, Caracciolo, Sokal and Sportiello presented, inter alia, an algebraic/combinatorial proof for Cayley's identity. The purpose of the present paper is to give a "purely combinatorial" proof for this identity; i.e., a proof…
We construct a homotopy initial functor from the partition complex of a finite set $A$ to a category of trees with leaves labelled by $A$. As an application, this provides an equivalence between different bar constructions of an operad. In…
Although false for general graphs, this note gives an elementary proof of the bunkbed conjecture for any acyclic graph. The argument is short and self-contained, and may be of educational interest.
Recently, Andrews and El Bachraoui considered the number of integer partitions whose smallest part is repeated exactly $k$ times and the remaining parts are not repeated. They presented several interesting results and posed questions…
Recently, Andrews and El Bachraoui (2024) proved three very interesting $q$-series identities, from which three simple looking identities involving certain restricted partitions into distinct even parts and $4$-regular partitions follow. In…
We study rational homology groups of one-point compactifications of spaces of complex monic polynomials with multiple roots. These spaces are indexed by number partitions. A standard reformulation in terms of quotients of orbit arrangements…
We give an explicit combinatorial proof of a weighted version of strong log-concavity for the generating polynomial of increasing spanning forests of a finite simple graph equipped with a total ordering of the vertices. In contrast to…
In this note, by counting some colored plane trees we obtain several binomial identities. These identities can be viewed as specific evaluations of certain generalizations of the Narayana polynomials. As consequences, it provides…
Binary relations derived from labeled rooted trees play an import role in mathematical biology as formal models of evolutionary relationships. The (symmetrized) Fitch relation formalizes xenology as the pairs of genes separated by at least…
We present a surprisingly new connection between two well-studied combinatorial classes: rooted connected chord diagrams on one hand, and rooted bridgeless combinatorial maps on the other hand. We describe a bijection between these two…
The matrices of spanning rooted forests are studied as a tool for analysing the structure of networks and measuring their properties. The problems of revealing the basic bicomponents, measuring vertex proximity, and ranking from preference…
The class of self-nested trees presents remarkable compression properties because of the systematic repetition of subtrees in their structure. In this paper, we provide a better combinatorial characterization of this specific family of…
It is well known that the number of partitions into distinct even parts equals the number of $4$-regular partitions. In this paper we prove identities relating certain restricted partitions into distinct even parts with restricted…
We study the problem of decomposing (clustering) a tree with respect to costs attributed to pairs of nodes, so as to minimize the sum of costs for those pairs of nodes that are in the same component (cluster). For the general case and for…
In this short note, we find the number of forests of chord diagrams with a given number of trees and a given number of chords.
We enumerate injectively $k$-colored rooted forests with a given number of vertices of each color and a given sequence of root colors. We obtain from this result some new multi-parameter distributions of Fuss-Catalan numbers. As an…
A preferential arrangement of a finite set is an ordered partition. Associated with each such ordered partition is a chain of subsets or blocks endowed with a linear order. The chain may be split into sections by the introduction of a…
The aim of this short note is to show how can be derived from the properties of fundamental interpolation polynomials some nice identities.