Geodesic
The descent set and connectivity set of a permutation
Journal of integer sequences, Tome 8 (2005) no. 3
The descent set $D(w)$ of a permutation $w$ of 1,2,$\dots ,n$ is a standard and well-studied statistic. We introduce a new statistic, the connectivity set $C(w)$, and show that it is a kind of dual object to $D(w)$. The duality is stated in terms of the inverse of a matrix that records the joint distribution of $D(w)$ and $C(w)$. We also give a variation involving permutations of a multiset and a $q$-analogue that keeps track of the number of inversions of $w$.
Classification : 05A05
Keywords: descent set, connected permutation, connectivity set 
@article{JIS_2005__8_3_a0,
     author = {Stanley,  Richard P.},
     title = {The descent set and connectivity set of a permutation},
     journal = {Journal of integer sequences},
     year = {2005},
     volume = {8},
     number = {3},
     zbl = {1101.05002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2005__8_3_a0/}
}
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Stanley,  Richard P. The descent set and connectivity set of a permutation. Journal of integer sequences, Tome 8 (2005) no. 3. http://geodesic.mathdoc.fr/item/JIS_2005__8_3_a0/