2,854 research outputs found
On a conjecture of Wilf
Let n and k be natural numbers and let S(n,k) denote the Stirling numbers of
the second kind. It is a conjecture of Wilf that the alternating sum
\sum_{j=0}^{n} (-1)^{j} S(n,j) is nonzero for all n>2. We prove this conjecture
for all n not congruent to 2 and not congruent to 2944838 modulo 3145728 and
discuss applications of this result to graph theory, multiplicative partition
functions, and the irrationality of p-adic series.Comment: 18 pages, final version, accepted for publication in the Journal of
Combinatorial Theory, Series
The largest singletons of set partitions
Recently, Deutsch and Elizalde studied the largest and the smallest fixed
points of permutations. Motivated by their work, we consider the analogous
problems in set partitions. Let denote the number of partitions of
with the largest singleton for .
In this paper, several explicit formulas for , involving a
Dobinski-type analog, are obtained by algebraic and combinatorial methods, many
combinatorial identities involving and Bell numbers are presented by
operator methods, and congruence properties of are also investigated.
It will been showed that the sequences and
(mod ) are periodic for any prime , and contain a
string of consecutive zeroes. Moreover their minimum periods are
conjectured to be for any prime .Comment: 14page
The Zagier polynomials. Part II: Arithmetic properties of coefficients
The modified Bernoulli numbers \begin{equation*} B_{n}^{*} = \sum_{r=0}^{n}
\binom{n+r}{2r} \frac{B_{r}}{n+r}, \quad n > 0 \end{equation*} introduced by D.
Zagier in 1998 were recently extended to the polynomial case by replacing
by the Bernoulli polynomials . Arithmetic properties of the
coefficients of these polynomials are established. In particular, the 2-adic
valuation of the modified Bernoulli numbers is determined. A variety of
analytic, umbral, and asymptotic methods is used to analyze these polynomials
-regularity of the -adic valuation of the Fibonacci sequence
We show that the -adic valuation of the sequence of Fibonacci numbers is a
-regular sequence for every prime . For , we determine that
the rank of this sequence is , where is the
restricted period length of the Fibonacci sequence modulo .Comment: 7 pages; publication versio
On a curious property of Bell numbers
In this paper we derive congruences expressing Bell numbers and derangement
numbers in terms of each other modulo any prime.Comment: 6 page
The Distance Geometry of Music
We demonstrate relationships between the classic Euclidean algorithm and many
other fields of study, particularly in the context of music and distance
geometry. Specifically, we show how the structure of the Euclidean algorithm
defines a family of rhythms which encompass over forty timelines
(\emph{ostinatos}) from traditional world music. We prove that these
\emph{Euclidean rhythms} have the mathematical property that their onset
patterns are distributed as evenly as possible: they maximize the sum of the
Euclidean distances between all pairs of onsets, viewing onsets as points on a
circle. Indeed, Euclidean rhythms are the unique rhythms that maximize this
notion of \emph{evenness}. We also show that essentially all Euclidean rhythms
are \emph{deep}: each distinct distance between onsets occurs with a unique
multiplicity, and these multiplicies form an interval . Finally,
we characterize all deep rhythms, showing that they form a subclass of
generated rhythms, which in turn proves a useful property called shelling. All
of our results for musical rhythms apply equally well to musical scales. In
addition, many of the problems we explore are interesting in their own right as
distance geometry problems on the circle; some of the same problems were
explored by Erd\H{o}s in the plane.Comment: This is the full version of the paper: "The distance geometry of deep
rhythms and scales." 17th Canadian Conference on Computational Geometry (CCCG
'05), University of Windsor, Canada, 200
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