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Miklos Bona

Publications and source records attributed to Miklos Bona.

At least 19 recordsLinked to original sources
arXiv2026

On the intersection of pairs of trees

We consider the number of common edges in two independent random spanning trees of a graph $G$. For complete graphs $K_n$, we give a new proof of the fact, originally obtained by Moon, that the distribution converges to a Poisson distribution with expected value $2$. This is applied to show a Poisson limit law for the number of common edges in two independent random spanning trees of an Erdős--Rényi random graph $G(n,p)$ for constant~$p$, as well as a central limit theorem in the case where $p\to 0$ and $p\geq n^{-2/3+\varepsilon}$. We also use the same method to prove an analogous result for complete multipartite graphs.

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arXiv2026

Stack-sorting preimages and 0-1-trees

We define a class of partially labeled trees and use them to find simple proofs for two recent enumeration results of Colin Defant concerning stack-sorting preimages of permutation classes.

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arXiv2026

Boolean-Narayana numbers

We introduce a refinement of Boolean-Catalan numbers and call them Boolean-Narayana numbers. We provide an explicit formula for these numbers, and prove unimodality, log-concavity, and real-roots-only results for their sequences. We also prove a three-term recurrence relation for their generating polynomials.

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arXiv2024

Counting pairs of cycles whose product is a permutation with restricted cycle lengths

We find exact and asymptotic formulas for the number of pairs $(p,q)$ of $N$-cycles such that the all cycles of the product $p\cdot q$ have lengths from a given integer set. We then apply these results to prove a surprisingly high lower bound for the number of permutations whose block transposition distance from the identity is at least $(n+1)/2$.

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arXiv2024

The likely maximum size of twin subtrees in a large random tree

We call a pair of vertex-disjoint, induced subtrees of a rooted trees twins if they have the same counts of vertices by out-degrees. The likely maximum size of twins in a uniformly random, rooted Cayley tree of size $n\to\infty$ is studied. It is shown that the expected number of twins of size $(2+δ)\sqrt{\log n\cdot\log\log n}$ approaches zero, while the expected number of twins of size $(2-δ)\sqrt{\log n\cdot\log\log n}$ approaches infinity.

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arXiv2019

Cyclic permutations avoiding pairs of patterns of length three

We complete the enumeration of cyclic permutations avoiding two patterns of length three each by providing explicit formulas for all but one of the pairs for which no such formulas were known. The pair $(123,231)$ proves to be the most difficult of these pairs. We also prove a lower bound for the growth rate of the number of cyclic permutations that avoid a single pattern $q$, where $q$ is an element of a certain infinite family of patterns.

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arXiv2019

Pattern avoidance in permutations and their squares

We study permutations $p$ such that both $p$ and $p^2$ avoid a given pattern $q$. We obtain a generating function for the case of $q=312$ (equivalently, $q=231$), we prove that if $q$ is monotone increasing, then above a certain length, there are no such permutations, and we prove an upper bound for $q=321$. We also present some intriguing questions in the case of $q=132$.

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arXiv2019

An Involution on Involutions and a Generalization of Layered Permutations

Taking transposes of Standard Young Tableaux defines a natural involution on the set $I(n)$ of involutions of length $n$ via the the Robinson-Schensted correspondence. In some cases, this involution can be defined without resorting to the Robinson-Schensted correspondence. As a byproduct, we get an interesting generalization of layered permutations.

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arXiv2017

Balanced vertices in labeled rooted trees

In a rooted tree, we call a vertex {\em balanced} if it is at equal distance from all its descendant leaves. We count balanced vertices in three different tree varieties. For decreasing binary trees, we can prove that the probability that a vertex chosen uniformly at random from the set of all trees of a given size is balanced is monotone decreasing.

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arXiv2016

On the cycle structure of the product of random maximal cycles

The subject of this paper is the cycle structure of the random permutation $σ$ of $[N]$, which is the product of $k$ independent random cycles of maximal length $N$. We use the character-based Fourier transform to study the number of cycles of $σ$ and also the distribution of the elements of the subset $[\ell]$ among the cycles of $σ$.

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arXiv2016

Symmetry in Sphere-based Assembly Configuration Spaces

Many remarkably robust, rapid and spontaneous self-assembly phenomena in nature can be modeled geometrically starting from a collection of rigid bunches of spheres. This paper highlights the role of symmetry in sphere-based assembly processes. Since spheres within bunches could be identical and bunches could be identical as well, the underlying symmetry groups could be of large order that grows with the number of participating spheres and bunches. Thus, understanding symmetries and associated isomorphism classes of microstates correspond to various types of macrostates can significantly reduce the complexity of computing entropy and free energy, as well as paths and kinetics, in high dimensional configuration spaces. In addition, a precise understanding of symmetries is crucial for giving provable guarantees of algorithmic accuracy and efficiency in such computations. In particular, this may aid in predicting crucial assembly-driving interactions. This is a primarily expository paper that develops a novel, original framework for dealing with symmetries in configuration spaces of assembling spheres with the following goals. (1) We give new, formal definitions of various concepts relevant to sphere-based assembly that occur in previous work, and in turn, formal definitions of their relevant symmetry groups leading to the main theorem concerning their symmetries. These previously developed concepts include, for example, (a) assembly configuration spaces, (b) stratification of assembly configuration space into regions defined by active constraint graphs, (c) paths through the configurational regions, and (d) coarse assembly pathways. (2) We demonstrate the new symmetry concepts to compute sizes and numbers of orbits in two example settings appearing in previous work. (3) We give formal statements of a variety of open problems and challenges using the new conceptual definitions.

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arXiv2015

A Bijective Proof of and Identity Extending a Classic Result of Hajos

We provide bijective proofs of two classic identities that are very simple to prove using generating functions, but surprisingly difficult to prove combinatorially. The problem of finding a bijective proof for the first identity was first raised in the 1930s. The second, more involved identity takes the first one a step further.

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