207 research outputs found
The Star of David Rule
In this note, a new concept called {\em -matrix} is proposed, which is
an infinite lower triangular matrix obeying the generalized rule of David star.
Some basic properties of -matrices are discussed and two conjectures on
-matrices are presented, one of which states that if a matrix is a
-matrix, then so is its matrix inverse (if exists).Comment: 7 pages, 1 figur
Two kinds of hook length formulas for complete -ary trees
In this paper, we define two kinds of hook length for internal vertices of
complete -ary trees, and deduce their corresponding hook length formulas,
which generalize the main results obtained by Du and Liu.Comment: 6 pages, 1 figure. To appear in Discrete Mathematic
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 largest singletons in weighted set partitions and its applications
Recently, Deutsch and Elizalde studied the largest and the smallest fixed
points of permutations. Motivated by their work, we consider the analogous
problems in weighted set partitions. Let denote the total
weight of partitions on with the largest singleton . In this
paper, explicit formulas for and many combinatorial
identities involving are obtained by umbral operators and
combinatorial methods. As applications, we investigate three special cases such
as permutations, involutions and labeled forests. Particularly in the
permutation case, we derive a surprising identity analogous to the Riordan
identity related to tree enumerations, namely, \begin{eqnarray*}
\sum_{k=0}^{n}\binom{n}{k}D_{k+1}(n+1)^{n-k} &=& n^{n+1}, \end{eqnarray*} where
is the -th derangement number or the number of permutations of
with no fixed points.Comment: 15page
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