136 research outputs found
Parking functions on toppling matrices
Let be an integer -matrix which satisfies the
conditions: , and
there exists a vector such that . Here the notation means that for all , and
means that for every . Let
be the set of vectors such that and
. In this paper, -parking functions are
defined for any . It is proved that the set of
-parking functions is independent of for any . For this reason, -parking
functions are simply called -parking functions. It is shown that the
number of -parking functions is less than or equal to the determinant
of . Moreover, the definition of -recurrent
configurations are given for any . It is proved
that the set of -recurrent configurations is independent of
for any . Hence, -recurrent configurations are simply called -recurrent
configurations. It is obtained that the number of -recurrent
configurations is larger than or equal to the determinant of . A simple
bijection from -parking functions to -recurrent configurations
is established. It follows from this bijection that the number of
-parking functions and the number of -recurrent configurations
are both equal to the determinant of
On various restricted sumsets
For finite subsets A_1,...,A_n of a field, their sumset is given by
{a_1+...+a_n: a_1 in A_1,...,a_n in A_n}. In this paper we study various
restricted sumsets of A_1,...,A_n with restrictions of the following forms:
a_i-a_j not in S_{ij}, or alpha_ia_i not=alpha_ja_j, or a_i+b_i not=a_j+b_j
(mod m_{ij}). Furthermore, we gain an insight into relations among recent
results on this area obtained in quite different ways.Comment: 11 pages; final version for J. Number Theor
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