Geodesic
Partition lattice $q$-analogs related to $q$-Stirling numbers
Journal of Algebraic Combinatorics, Tome 3 (1994) no. 3, pp. 261-283.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We construct a family of partially ordered sets (posets) that are $q$-analogs of the set partition lattice. They are different from the $q$-analogs proposed by Dowling [5]. One of the important features of these posets is that their Whitney numbers of the first and second kind are just the $q$-Stirling numbers of the first and second kind, respectively. One member of this family [4] can be constructed using an interpretation of Milne [9] for $S[ n, k]$ as sequences of lines in a vector space over the Galois field $F _{q}$. Another member is constructed so as to mirror the partial order in the subspace lattice.
Keywords: set partition lattice, vector space over a finite field, $q$-Stirling number 
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     author = {Bennett, Curtis and Dempsey, Kathy J. and Sagan, Bruce E.},
     title = {Partition lattice $q$-analogs related to $q${-Stirling} numbers},
     journal = {Journal of Algebraic Combinatorics},
     pages = {261--283},
     publisher = {mathdoc},
     volume = {3},
     number = {3},
     year = {1994},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_1994__3_3_a3/}
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Bennett, Curtis; Dempsey, Kathy J.; Sagan, Bruce E. Partition lattice $q$-analogs related to $q$-Stirling numbers. Journal of Algebraic Combinatorics, Tome 3 (1994) no. 3, pp. 261-283. http://geodesic.mathdoc.fr/item/JAC_1994__3_3_a3/