Geodesic
Geometric constructions of higher Bruhat orders and $M$-Morsifications
Izvestiya. Mathematics , Tome 60 (1996) no. 6, pp. 1183-1192.

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Higher Bruhat orders were introduced by Manin and Schechtman in the course of studying multi-dimensional generalizations of the Yang–Baxter equations. In this paper we present a problem from real singularity theory which generalizes Arnol'ds snake calculus (a coding of the connected components of the space of very nice $M$-Morsifications of a singularity $A_n$) and in which the role of updown permutations is played by their higher analogues – elements of special form in higher Bruhat orders. 
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G. G. Ilyuta. Geometric constructions of higher Bruhat orders and $M$-Morsifications. Izvestiya. Mathematics , Tome 60 (1996) no. 6, pp. 1183-1192. http://geodesic.mathdoc.fr/item/IM2_1996_60_6_a2/

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