Geodesic
Counting biorders
Journal of integer sequences, Tome 6 (2003) no. 4
Biorders were introduced first as Guttman scales and then as Ferrers relations. They are now well recognized in combinatorics and its applications. However, it seems that no procedure besides plain enumeration was made available for obtaining the number of biorders from an $m$-element set to an $n$-element set. We establish first a double-recurrence formula for computing this number, and then two explicit formulas involving Stirling numbers of the second kind. Our methods do not seem to extend to other, similar structures. For instance, interval orders on a finite set are exactly the irreflexive biorders on that set. To our knowledge, no direct formula is available for deriving their number.
Classification : 05A15, 06A06
Keywords: biorder, Stirling number of the second kind 
@article{JIS_2003__6_4_a6,
     author = {Christophe,  Julie and Doignon,  Jean-Paul and Fiorini,  Samuel},
     title = {Counting biorders},
     journal = {Journal of integer sequences},
     year = {2003},
     volume = {6},
     number = {4},
     zbl = {1064.05005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2003__6_4_a6/}
}
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Christophe,  Julie; Doignon,  Jean-Paul; Fiorini,  Samuel. Counting biorders. Journal of integer sequences, Tome 6 (2003) no. 4. http://geodesic.mathdoc.fr/item/JIS_2003__6_4_a6/