  The introduction of bars in-between blocks of an ordered set
partition(preferential arrangement) results in a barred ordered set
partition(barred preferential arrange- ment). Having the restriction that some
blocks of barred preferential arrangements to have a maximum of one block
results in restricted barred preferential arrange- ments. In this study we
establish relations between number of restricted barred preferential
arrangements, multi-poly-Bernoulli numbers and numbers related to
multi-poly-Bernoulli numbers. We prove a periodicity property satisfied by
multi- poly-Bernoulli numbers having negative index, number of restricted
barred prefer- ential arrangements and numbers related to multi-poly-Bernoulli
numbers having negative index.
