An
-descent in a permutation is a pair of adjacent elements
such that the first element is from
, the second element is from
, and the first element is greater than the second one. An
-adjacency in a permutation is a pair of adjacent elements
such that the first one is from
and the second one is from
.
An
-place-value pair in a permutation is an element
in
position
, such that
is in
and
is in
. It turns
out, that for certain choices of
and
some of the three
statistics above become equidistributed. Moreover, it is easy to
derive the distribution formula for
-place-value pairs thus
providing distribution for other statistics under consideration too.
This generalizes some results in the literature. As a result of our
considerations, we get combinatorial proofs of several remarkable
identities. We also conjecture existence of a bijection between two
objects in question preserving a certain statistic.