362 research outputs found
Lucas' theorem: its generalizations, extensions and applications (1878--2014)
In 1878 \'E. Lucas proved a remarkable result which provides a simple way to
compute the binomial coefficient modulo a prime in terms of
the binomial coefficients of the base- digits of and : {\it If is
a prime, and are the
-adic expansions of nonnegative integers and , then
\begin{equation*} {n\choose m}\equiv \prod_{i=0}^{s}{n_i\choose m_i}\pmod{p}.
\end{equation*}}
The above congruence, the so-called {\it Lucas' theorem} (or {\it Theorem of
Lucas}), plays an important role in Number Theory and Combinatorics. In this
article, consisting of six sections, we provide a historical survey of Lucas
type congruences, generalizations of Lucas' theorem modulo prime powers, Lucas
like theorems for some generalized binomial coefficients, and some their
applications.
In Section 1 we present the fundamental congruences modulo a prime including
the famous Lucas' theorem. In Section 2 we mention several known proofs and
some consequences of Lucas' theorem. In Section 3 we present a number of
extensions and variations of Lucas' theorem modulo prime powers. In Section 4
we consider the notions of the Lucas property and the double Lucas property,
where we also present numerous integer sequences satisfying one of these
properties or a certain Lucas type congruence. In Section 5 we collect several
known Lucas type congruences for some generalized binomial coefficients. In
particular, this concerns the Fibonomial coefficients, the Lucas -nomial
coefficients, the Gaussian -nomial coefficients and their generalizations.
Finally, some applications of Lucas' theorem in Number Theory and Combinatorics
are given in Section 6.Comment: 51 pages; survey article on Lucas type congruences closely related to
Lucas' theore
Relations on words
In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our presentation tries to set up a unified framework with respect to a given binary relation.
In the second part, we mainly focus on abelian equivalence, -abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and -equivalence. In particular, some new refinements of abelian equivalence are introduced
On the topology of the permutation pattern poset
The set of all permutations, ordered by pattern containment, forms a poset.
This paper presents the first explicit major results on the topology of
intervals in this poset. We show that almost all (open) intervals in this poset
have a disconnected subinterval and are thus not shellable. Nevertheless, there
seem to be large classes of intervals that are shellable and thus have the
homotopy type of a wedge of spheres. We prove this to be the case for all
intervals of layered permutations that have no disconnected subintervals of
rank 3 or more. We also characterize in a simple way those intervals of layered
permutations that are disconnected. These results carry over to the poset of
generalized subword order when the ordering on the underlying alphabet is a
rooted forest. We conjecture that the same applies to intervals of separable
permutations, that is, that such an interval is shellable if and only if it has
no disconnected subinterval of rank 3 or more. We also present a simplified
version of the recursive formula for the M\"obius function of decomposable
permutations given by Burstein et al.Comment: 33 pages, 4 figures. Incorporates changes suggested by the referees;
new open problems in Subsection 9.4. To appear in JCT(A
The congruence of Wolstenholme and generalized binomial coefficients related to Lucas sequences
Using generalized binomial coefficients with respect to fundamental Lucas
sequences we establish congruences that generalize the classical congruence of
Wolstenholme and other related stronger congruences.Comment: 23 page
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