Geodesic
Width-$k$ generalizations of classical permutation statistics
Journal of integer sequences, Tome 20 (2017) no. 6.

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Summary: We introduce new natural generalizations of the classical descent and inversion statistics for permutations, called $width-k$ descents and $width-k$ inversions. These variations induce generalizations of the excedance and major statistics, providing a framework in which well-known equidistributivity results for classical statistics are paralleled. We explore additional relationships among the statistics providing specific formulas in certain special cases. Moreover, we explore the behavior of these width-$k$ statistics in the context of pattern avoidance.
Classification : 05A05, 05A15
Keywords: permutation statistics, pattern avoidance, generating function 
@article{JIS_2017__20_6_a2,
     author = {Davis, Robert},
     title = {Width-$k$ generalizations of classical permutation statistics},
     journal = {Journal of integer sequences},
     publisher = {mathdoc},
     volume = {20},
     number = {6},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2017__20_6_a2/}
}
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Davis, Robert. Width-$k$ generalizations of classical permutation statistics. Journal of integer sequences, Tome 20 (2017) no. 6. http://geodesic.mathdoc.fr/item/JIS_2017__20_6_a2/