arXiv subjects

Miklos Bona

Publications and source records attributed to Miklos Bona.

43 records · Page 3Linked to original sources

arXiv2006

On a Balanced Property of Compositions

Let $S$ be a finite set of positive integers with largest element $m$. Let us randomly select a composition $a$ of the integer $n$ with parts in $S$, and let $m(a)$ be the multiplicity of $m$ as a part of $a$. Let $0\leq r<q$ be integers, with $q\geq 2$, and let $p_{n,r}$ be the probability that $m(a)$ is congruent to $r$ modulo $q$. We show that if $S$ satisfies a certain simple condition, then $\lim_{n\to \infty} p_{n,r} =1/q$. In fact, we show that an obvious necessary condition on $S$ turns out to be sufficient.

math.CO↗

arXiv2006

On a balanced property of derangements

We prove an interesting fact describing the location of the roots of the generating polynomials of the numbers of derangements of length $n$, counted by their number of cycles. We then use this result to prove that if $k$ is the number of cycles of a randomly selected derangement of length $n$, then the probability that $k$ is congruent to a given $r$ modulo a given $q$ converges to $1/q$. Finally, we generalize our results to $a$-derangements, which are permutations in which each cycle is longer than $a$.

math.NA↗

arXiv2005

On divisibility of Narayana numbers by primes

Using Kummer's Theorem, we give a necessary and sufficient condition for a Narayana number to be divisible by a given prime. We use this to derive certain properties of the Narayana triangle.

math.CO↗

arXiv2004

The limit of a Stanley-Wilf sequence is not always rational, and layered patterns beat monotone patterns

We show the first known example for a pattern $q$ for which $\lim_{n\to \infty} \sqrt[n]{S_n(q)}$ is not an integer. We find the exact value of the limit and show that it is irrational. Then we generalize our results to an infinite sequence of patterns. Finally, we provide further generalizations that start explaining why certain patterns are easier to avoid than others. Finally, we show that if $q$ is a layered pattern of length $k$, then $L(q)\geq (k-1)^2$ holds.

math.CO↗

arXiv2001

Pattern frequency sequences and internal zeros

Consider the number of permutations in the symmetric group on n letters that contain c copies of a given pattern. As c varies (with n held fixed) these numbers form a sequence whose properties we study for the monotone patterns and the patterns 1, l, l-1, ..., 2. We show that, except for the patterns 1, 2 and 2, 1 where the sequence is well-known to be log concave, there are infinitely many n where the sequence has internal zeros.

math.CO↗

arXiv1999

Enumeration of m-ary cacti

The purpose of this paper is to enumerate various classes of cyclically colored m-gonal plane cacti, called m-ary cacti. This combinatorial problem is motivated by the topological classification of complex polynomials having at most m critical values, studied by Zvonkin and others. We obtain explicit formulae for both labelled and unlabelled m-ary cacti, according to i) the number of polygons, ii) the vertex-color distribution, iii) the vertex-degree distribution of each color. We also enumerate m-ary cacti according to the order of their automorphism group. Using a generalization of Otter's formula, we express the species of m-ary cacti in terms of rooted and of pointed cacti. A variant of the m-dimensional Lagrange inversion is then used to enumerate these structures. The method of Liskovets for the enumeration of unrooted planar maps can also be adapted to m-ary cacti.

math.CO↗

arXiv1998

A sefl-dual poset on objects counted by the Catalan numbers

We examine the poset $P$ of 132-avoiding $n$-permutations ordered by descents. We show that this poset is the "coarsening" of the well-studied poset $Q$ of noncrossing partitions . In other words, if $x<y$ in $Q$, then $f(y)<f(x)$ in $P$, where $f$ is the canonical bijection from the set of noncrossing partitions onto that of 132-avoiding permutations. This enables us to prove many properties of $P$.

math.CO↗