An
![$(X,Y)$](abs/img1.gif)
-descent in a permutation is a pair of adjacent elements
such that the first element is from
![$X$](abs/img2.gif)
, the second element is from
![$Y$](abs/img3.gif)
, and the first element is greater than the second one. An
![$(X,Y)$](abs/img1.gif)
-adjacency in a permutation is a pair of adjacent elements
such that the first one is from
![$X$](abs/img2.gif)
and the second one is from
![$Y$](abs/img3.gif)
.
An
![$(X,Y)$](abs/img1.gif)
-place-value pair in a permutation is an element
![$y$](abs/img4.gif)
in
position
![$x$](abs/img5.gif)
, such that
![$y$](abs/img4.gif)
is in
![$Y$](abs/img3.gif)
and
![$x$](abs/img5.gif)
is in
![$X$](abs/img2.gif)
. It turns
out, that for certain choices of
![$X$](abs/img2.gif)
and
![$Y$](abs/img3.gif)
some of the three
statistics above become equidistributed. Moreover, it is easy to
derive the distribution formula for
![$(X,Y)$](abs/img1.gif)
-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.