Geodesic
Equidistribution of descents, adjacent pairs, and place-value pairs on permutations
Journal of integer sequences, Tome 12 (2009) no. 5
An $(X,Y)$-descent in a permutation is a pair of adjacent elements such that the first element is from $X$, the second element is from $Y$, and the first element is greater than the second one. An $(X,Y)$-adjacency in a permutation is a pair of adjacent elements such that the first one is from $X$ and the second one is from $Y$. An $(X,Y)$-place-value pair in a permutation is an element $y$ in position $x$, such that $y$ is in $Y$ and $x$ is in $X$. It turns out, that for certain choices of $X$ and $Y$ some of the three statistics above become equidistributed. Moreover, it is easy to derive the distribution formula for $(X,Y)$-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.
Classification : 05A15, 05A19, 05A05
Keywords: descent, adjacent pair, place-value-pair, equidistribution, binomial identity 
@article{JIS_2009__12_5_a4,
     author = {Deutsch,  Emeric and Kitaev,  Sergey and Remmel,  Jeffrey},
     title = {Equidistribution of descents, adjacent pairs, and place-value pairs on permutations},
     journal = {Journal of integer sequences},
     year = {2009},
     volume = {12},
     number = {5},
     zbl = {1213.05004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2009__12_5_a4/}
}
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Deutsch,  Emeric; Kitaev,  Sergey; Remmel,  Jeffrey. Equidistribution of descents, adjacent pairs, and place-value pairs on permutations. Journal of integer sequences, Tome 12 (2009) no. 5. http://geodesic.mathdoc.fr/item/JIS_2009__12_5_a4/