Abstract
In order to study signed Eulerian numbers, we introduce permutations of a
particular type, called parity-alternate permutations, because they take even
and odd entries alternately. The objective of this paper is twofold. The first
is to derive several properties of those permutations, by subdividing them into
even and odd ones. The second is to discuss relationships between those and
signed Eulerian numbers. Divisibility properties by prime powers are also
deduced for signed Eulerian numbers and several related numbers.
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